"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { a: "a", annotation: "annotation", h3: "h3", h4: "h4", li: "li", math: "math", mfrac: "mfrac", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", mstyle: "mstyle", msub: "msub", msubsup: "msubsup", msup: "msup", mtable: "mtable", mtd: "mtd", mtr: "mtr", munder: "munder", munderover: "munderover", p: "p", path: "path", semantics: "semantics", span: "span", strong: "strong", svg: "svg", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsx(_components.p, { children: "A series is studied through its sequence of partial sums" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "s_n=\\sum_{k=0}^{n} a_k" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9535em", verticalAlign: "-1.3021em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8479em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.3021em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["so series convergence is a special case of ", _jsx(_components.a, { href: "/Math/Analysis+ETH/Sequence", className: "internal", "data-fs-resolved-file-path": "/Math/Analysis ETH/Sequence.md", children: "sequence convergence" }), "."] }), "\n", _jsxs(_components.p, { children: ["For worked examples, see ", _jsx(_components.a, { href: "/Math/Analysis+ETH/Convergence+test+examples", className: "internal", "data-fs-resolved-file-path": "/Math/Analysis ETH/Convergence test examples.md", children: "Convergence test examples" }), "."] }), "\n", _jsxs(_components.p, { children: ["For a series ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} a_n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), ":"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["it is ", _jsx(_components.strong, { children: "absolutely convergent" }), " if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} |a_n|" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mord", children: "∣" })] }) })] }), " converges"] }), "\n", _jsxs(_components.li, { children: ["it is ", _jsx(_components.strong, { children: "conditionally convergent" }), " if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} a_n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " converges, but ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} |a_n|" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mord", children: "∣" })] }) })] }), " does not"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "riemann-rearrangement-theorem", children: [_jsx(_components.a, { className: "heading-link", href: "#riemann-rearrangement-theorem", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Riemann Rearrangement Theorem"] }), "\n", _jsxs(_components.p, { children: ["If a real series is conditionally convergent, then for every ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "L" }), _jsx(_components.mo, { children: "∈" }), _jsx(_components.mi, { mathvariant: "double-struck", children: "R" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "L \\in \\mathbb{R}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7224em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6889em" } }), _jsx(_components.span, { className: "mord mathbb", children: "R" })] })] })] }), " there is a bijection ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "φ" }), _jsx(_components.mo, { children: ":" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "double-struck", children: "N" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "→" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "double-struck", children: "N" }), _jsx(_components.mn, { children: "0" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\varphi:\\mathbb{N}_0 \\to \\mathbb{N}_0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "φ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ":" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbb", children: "N" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbb", children: "N" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }), " such that"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "φ" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} a_{\\varphi(n)} = L" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9185em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3448em" }, children: _jsxs(_components.span, { style: { top: "-2.5198em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "φ" }), _jsx(_components.span, { className: "mopen mtight", children: "(" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mclose mtight", children: ")" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.3552em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] })] })] }) }), "\n", _jsx(_components.p, { children: "So a conditionally convergent series can be rearranged to converge to any prescribed real value. In contrast, absolutely convergent series keep the same sum under every rearrangement." }), "\n", _jsxs(_components.h3, { id: "leibniz-criterion", children: [_jsx(_components.a, { className: "heading-link", href: "#leibniz-criterion", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Leibniz Criterion"] }), "\n", _jsxs(_components.p, { children: ["If ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "∈" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "double-struck", children: "N" }), _jsx(_components.mn, { children: "0" })] })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(a_n)_{n\\in\\mathbb{N}_0}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0001em", verticalAlign: "-0.2501em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3322em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "∈" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathbb mtight", children: "N" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3173em" }, children: _jsxs(_components.span, { style: { top: "-2.357em", marginLeft: "0em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.143em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2501em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " is a monotone decreasing sequence of non-negative real numbers and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_n \\to 0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), ", then the alternating series"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} (-1)^n a_n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9185em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "\n", _jsx(_components.p, { children: "converges." }), "\n", _jsxs(_components.h3, { id: "cauchy-criterion", children: [_jsx(_components.a, { className: "heading-link", href: "#cauchy-criterion", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Cauchy Criterion"] }), "\n", _jsxs(_components.p, { children: ["The series ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} a_n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " converges if and only if for every ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ε" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\varepsilon > 0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ε" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), " there exists ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { children: "∈" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "double-struck", children: "N" }), _jsx(_components.mn, { children: "0" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "N \\in \\mathbb{N}_0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7224em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, children: "N" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathbb", children: "N" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }), " such that for all ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "≥" }), _jsx(_components.mi, { children: "N" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n \\ge m \\ge N" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7719em", verticalAlign: "-0.136em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, children: "N" })] })] })] }), ","] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "∣" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { fence: "true", children: "∣" })] }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "ε" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\left| \\sum_{k=m+1}^{n} a_k \\right| < \\varepsilon" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "3.1104em", verticalAlign: "-1.3604em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen", children: _jsx(_components.span, { className: "delimsizing mult", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "1.75em" }, children: _jsxs(_components.span, { style: { top: "-3.75em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "5em" } }), _jsx(_components.span, { style: { width: "0.333em", height: "3em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "0.333em", height: "3em", viewBox: "0 0 333 3000", children: _jsx(_components.path, { d: "M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1800 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1800 v585 h43z" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.25em" }, children: _jsx(_components.span, {}) }) })] }) }) }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8479em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.3604em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: _jsx(_components.span, { className: "delimsizing mult", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "1.75em" }, children: _jsxs(_components.span, { style: { top: "-3.75em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "5em" } }), _jsx(_components.span, { style: { width: "0.333em", height: "3em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "0.333em", height: "3em", viewBox: "0 0 333 3000", children: _jsx(_components.path, { d: "M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1800 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1800 v585 h43z" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.25em" }, children: _jsx(_components.span, {}) }) })] }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ε" })] })] })] }) }), "\n", _jsx(_components.p, { children: "So the tails of the series must become arbitrarily small." }), "\n", _jsxs(_components.h3, { id: "p-series-test", children: [_jsx(_components.a, { className: "heading-link", href: "#p-series-test", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "p-Series Test"] }), "\n", _jsx(_components.p, { children: "The series" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "p" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=1}^{\\infty} \\frac{1}{n^p}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9185em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) })] }) }), "\n", _jsxs(_components.p, { children: ["converges if and only if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p>1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Proof" }), ":"] }), "\n", _jsxs(_components.p, { children: ["If ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p>1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", split the series into dyadic blocks:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "p" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] })] }), _jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "p" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=1}^{\\infty} \\frac{1}{n^p}\n= 1 + \\sum_{m=0}^{\\infty}\\sum_{n=2^m}^{2^{m+1}-1}\\frac{1}{n^p}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9185em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "3.241em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.9739em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5935em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["For ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "<" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2^m \\le n < 2^{m+1}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8004em", verticalAlign: "-0.136em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] })] })] })] }), ", we have"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "p" })] })] }), _jsx(_components.mo, { children: "≤" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "p" })] })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{1}{n^p} \\le \\frac{1}{(2^m)^p}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.2574em", verticalAlign: "-0.936em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.936em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["and there are ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2^m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6644em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] })] }) })] }), " terms in this block. Hence"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] })] }), _jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "p" })] })] }), _jsx(_components.mo, { children: "≤" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mo, { children: "⋅" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "p" })] })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=2^m}^{2^{m+1}-1}\\frac{1}{n^p}\n\\le 2^m \\cdot \\frac{1}{(2^m)^p}\n= 2^{-m(p-1)}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "3.241em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.9739em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5935em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7144em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "⋅" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.2574em", verticalAlign: "-0.936em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.936em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.938em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.938em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mopen mtight", children: "(" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" }), _jsx(_components.span, { className: "mclose mtight", children: ")" })] }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "Therefore" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "p" })] })] }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=1}^{\\infty} \\frac{1}{n^p}\n\\le 1 + \\sum_{m=0}^{\\infty} 2^{-m(p-1)}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9185em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5904em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9185em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.938em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mopen mtight", children: "(" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "p" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" }), _jsx(_components.span, { className: "mclose mtight", children: ")" })] }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["and the right-hand side is a convergent geometric series because ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p-1>0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6835em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), "."] }), "\n", _jsxs(_components.p, { children: ["If ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p=1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", then for ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mo, { children: "≤" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "<" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2^m \\le n < 2^{m+1}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8004em", verticalAlign: "-0.136em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≤" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] })] })] })] }), " we have"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { children: "≥" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{1}{n} \\ge \\frac{1}{2^{m+1}}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7401em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "so" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] })] }), _jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { children: "≥" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { children: "m" })] }), _jsx(_components.mo, { children: "⋅" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "2" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=2^m}^{2^{m+1}-1}\\frac{1}{n}\n\\ge 2^m \\cdot \\frac{1}{2^{m+1}}\n= \\frac{1}{2}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "3.241em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.9739em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.5935em" }, children: _jsxs(_components.span, { style: { top: "-2.786em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7144em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "⋅" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7401em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "m" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "2" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["Thus each dyadic block contributes at least ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{1}{2}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1901em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8451em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) })] }), ", so the harmonic series diverges."] }), "\n", _jsxs(_components.p, { children: ["If ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_n,b_n>0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), " for all sufficiently large ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), ". Assume"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.mfrac, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "c" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\lim_{n\\to\\infty}\\frac{a_n}{b_n}=c" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.9436em", verticalAlign: "-0.836em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.4em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop", children: "lim" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.1076em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.836em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" })] })] })] }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["If ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\rho > 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", the series diverges"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "ratio-test", children: [_jsx(_components.a, { className: "heading-link", href: "#ratio-test", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Ratio Test"] }), "\n", _jsxs(_components.p, { children: ["If ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { mathvariant: "normal", children: "≠" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_n \\neq 0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mrel", children: [_jsx(_components.span, { className: "mrel", children: _jsx(_components.span, { className: "mord vbox", children: _jsx(_components.span, { className: "thinbox", children: _jsxs(_components.span, { className: "rlap", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "inner", children: _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mrel", children: "" }) }) }), _jsx(_components.span, { className: "fix" })] }) }) }) }), _jsx(_components.span, { className: "mspace nobreak" }), _jsx(_components.span, { className: "mrel", children: "=" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), " for all ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), " and"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "∣" }), _jsxs(_components.mfrac, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { fence: "true", children: "∣" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\rho = \\lim_{n\\to\\infty} \\left| \\frac{a_{n+1}}{a_n} \\right|" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.4em", verticalAlign: "-0.95em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.4em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop", children: "lim" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen", children: _jsx(_components.span, { className: "delimsizing mult", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "1.45em" }, children: _jsxs(_components.span, { style: { top: "-3.45em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "4.4em" } }), _jsx(_components.span, { style: { width: "0.333em", height: "2.4em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "0.333em", height: "2.4em", viewBox: "0 0 333 2400", children: _jsx(_components.path, { d: "M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.95em" }, children: _jsx(_components.span, {}) }) })] }) }) }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.1076em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2083em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.836em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mclose", children: _jsx(_components.span, { className: "delimsizing mult", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "1.45em" }, children: _jsxs(_components.span, { style: { top: "-3.45em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "4.4em" } }), _jsx(_components.span, { style: { width: "0.333em", height: "2.4em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "0.333em", height: "2.4em", viewBox: "0 0 333 2400", children: _jsx(_components.path, { d: "M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.95em" }, children: _jsx(_components.span, {}) }) })] }) }) })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "then:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\rho < 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", the series ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} a_n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " converges absolutely"] }), "\n", _jsxs(_components.li, { children: ["if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\rho > 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", the series diverges"] }), "\n"] }), "\n", _jsxs(_components.h3, { id: "cauchy-product", children: [_jsx(_components.a, { className: "heading-link", href: "#cauchy-product", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Cauchy Product"] }), "\n", _jsxs(_components.p, { children: ["If ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} a_n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=0}^{\\infty} b_n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.104em", verticalAlign: "-0.2997em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " are absolutely convergent, then"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { fence: "true", children: ")" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { fence: "true", children: ")" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "k" })] })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { fence: "true", children: ")" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\left( \\sum_{n=0}^{\\infty} a_n \\right)\\left( \\sum_{n=0}^{\\infty} b_n \\right)\n= \\sum_{n=0}^{\\infty} \\left( \\sum_{k=0}^{n} a_{n-k} b_k \\right)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "3.0171em", verticalAlign: "-1.2671em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: "(" }) }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: ")" }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: "(" }) }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: ")" }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "3.0521em", verticalAlign: "-1.3021em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: "(" }) }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8479em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.3021em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2083em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: ")" }) })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "and the series on the right also converges absolutely." }), "\n", _jsxs(_components.h3, { id: "power-series", children: [_jsx(_components.a, { className: "heading-link", href: "#power-series", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Power Series"] }), "\n", _jsxs(_components.p, { children: ["A power series centered at ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }), " has the form"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "k" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "a" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{k=0}^{\\infty} c_k (x-a)^k" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9535em", verticalAlign: "-1.3021em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8479em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "0" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.3021em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1491em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8991em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["It has a radius of convergence ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "R" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "R" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0077em" }, children: "R" })] }) })] }), " such that:"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["it converges for ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "R" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "|x-a| < R" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0077em" }, children: "R" })] })] })] })] }), "\n", _jsxs(_components.li, { children: ["it diverges for ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mi, { children: "R" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "|x-a| > R" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0077em" }, children: "R" })] })] })] })] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["The interval ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "R" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "R" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(a-R, a+R)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0077em" }, children: "R" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0077em" }, children: "R" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), " is called the interval of convergence. If"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "lim sup" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "n" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\rho = \\limsup_{n\\to\\infty} |c_n|^{1/n}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.8324em", verticalAlign: "-0.8944em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.2056em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mord mathrm", children: "lim" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathrm", children: "sup" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.938em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1/" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "then" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "R" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "{" }), _jsxs(_components.mtable, { rowspacing: "0.36em", columnalign: "left left", columnspacing: "1em", children: [_jsxs(_components.mtr, { children: [_jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }) }) })] }), _jsxs(_components.mtr, { children: [_jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "ρ" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }) }) })] }), _jsxs(_components.mtr, { children: [_jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∞" }), _jsx(_components.mo, { separator: "true", children: "," })] }) }) }), _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }) }) })] })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "R =\n\\begin{cases}\n0, & \\rho = \\infty \\\\\n\\rho^{-1}, & 0 < \\rho < \\infty \\\\\n\\infty, & \\rho = 0\n\\end{cases}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0077em" }, children: "R" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "4.32em", verticalAlign: "-1.91em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen", children: _jsx(_components.span, { className: "delimsizing mult", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "2.35em" }, children: [_jsxs(_components.span, { style: { top: "-2.2em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.15em" } }), _jsx(_components.span, { className: "delimsizinginner delim-size4", children: _jsx(_components.span, { children: "⎩" }) })] }), _jsxs(_components.span, { style: { top: "-2.192em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.15em" } }), _jsx(_components.span, { style: { height: "0.316em", width: "0.8889em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "0.8889em", height: "0.316em", style: { width: "0.8889em" }, viewBox: "0 0 888.89 316", preserveAspectRatio: "xMinYMin", children: _jsx(_components.path, { d: "M384 0 H504 V316 H384z M384 0 H504 V316 H384z" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.15em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.15em" } }), _jsx(_components.span, { className: "delimsizinginner delim-size4", children: _jsx(_components.span, { children: "⎨" }) })] }), _jsxs(_components.span, { style: { top: "-4.292em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.15em" } }), _jsx(_components.span, { style: { height: "0.316em", width: "0.8889em" }, children: _jsx(_components.svg, { xmlns: "http://www.w3.org/2000/svg", width: "0.8889em", height: "0.316em", style: { width: "0.8889em" }, viewBox: "0 0 888.89 316", preserveAspectRatio: "xMinYMin", children: _jsx(_components.path, { d: "M384 0 H504 V316 H384z M384 0 H504 V316 H384z" }) }) })] }), _jsxs(_components.span, { style: { top: "-4.6em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.15em" } }), _jsx(_components.span, { className: "delimsizinginner delim-size4", children: _jsx(_components.span, { children: "⎧" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.85em" }, children: _jsx(_components.span, {}) }) })] }) }) }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mtable", children: [_jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "2.41em" }, children: [_jsxs(_components.span, { style: { top: "-4.41em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," })] })] }), _jsxs(_components.span, { style: { top: "-2.97em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mpunct", children: "," })] })] }), _jsxs(_components.span, { style: { top: "-1.53em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "∞" }), _jsx(_components.span, { className: "mpunct", children: "," })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.91em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "arraycolsep", style: { width: "1em" } }), _jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "2.41em" }, children: [_jsxs(_components.span, { style: { top: "-4.41em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord", children: "∞" })] })] }), _jsxs(_components.span, { style: { top: "-2.97em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord", children: "∞" })] })] }), _jsxs(_components.span, { style: { top: "-1.53em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.91em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["when using root test on power series: ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "a" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.mo, { children: "⟺" }), _jsxs(_components.mfrac, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "k" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "a" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_{k}=c_{k}(x-a)^{k}\\Longleftrightarrow \\frac{a_{k}}{c_{k}}=(x-a)^{k}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0991em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "⟺" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1681em", verticalAlign: "-0.4509em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.7173em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "c" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3448em" }, children: _jsxs(_components.span, { style: { top: "-2.3488em", marginLeft: "0em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1512em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4159em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3448em" }, children: _jsxs(_components.span, { style: { top: "-2.3488em", marginLeft: "0em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1512em" }, children: _jsx(_components.span, {}) }) })] }) })] }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.4509em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0991em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0315em" }, children: "k" }) }) })] }) }) }) }) })] })] })] })] })] }), "\n", _jsxs(_components.h4, { id: "example", children: [_jsx(_components.a, { className: "heading-link", href: "#example", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "Example"] }), "\n", _jsx(_components.p, { children: "Consider" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munderover, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=1}^{\\infty} (-1)^{n+1}\\frac{1}{n}x^n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.9185em", verticalAlign: "-1.2671em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.6514em" }, children: [_jsxs(_components.span, { style: { top: "-1.8829em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop op-symbol large-op", children: "∑" }) })] }), _jsxs(_components.span, { style: { top: "-4.3em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.05em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.2671em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }) }) }) })] })] }) })] }) }), "\n", _jsxs(_components.p, { children: ["This is a power series centered at ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a=0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }), " with coefficients"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "c_n = (-1)^{n+1}\\frac{1}{n}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "Hence" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { fence: "true", children: ")" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "|c_n|^{1/n} = \\left(\\frac{1}{n}\\right)^{1/n} \\to 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.188em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.938em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1/" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.6779em", verticalAlign: "-0.95em" } }), _jsxs(_components.span, { className: "minner", children: [_jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size3", children: "(" }) }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size3", children: ")" }) })] }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.7279em" }, children: _jsxs(_components.span, { style: { top: "-3.9029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1/" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) }), "\n", _jsx(_components.p, { children: "so" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "ρ" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "lim sup" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\rho = \\limsup_{n\\to\\infty} |c_n|^{1/n} = 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.8324em", verticalAlign: "-0.8944em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.2056em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mord mathrm", children: "lim" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathrm", children: "sup" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "c" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.938em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1/" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) }), "\n", _jsx(_components.p, { children: "and therefore" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "R" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "ρ" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "R = \\rho^{-1} = 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0077em" }, children: "R" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0585em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "ρ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) }), "\n", _jsx(_components.p, { children: "For the terms" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a_n(x) = (-1)^{n+1}\\frac{1}{n}x^n" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "the root test gives" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "lim sup" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "lim sup" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.msup, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "x" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { fence: "true", children: ")" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\limsup_{n\\to\\infty} |a_n(x)|^{1/n}\n= \\limsup_{n\\to\\infty} \\left(\\frac{1}{n}|x|^n\\right)^{1/n}\n= |x|" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.8324em", verticalAlign: "-0.8944em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.2056em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mord mathrm", children: "lim" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathrm", children: "sup" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.938em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1/" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.6779em", verticalAlign: "-0.95em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.2056em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mord mathrm", children: "lim" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathrm", children: "sup" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "minner", children: [_jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size3", children: "(" }) }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size3", children: ")" }) })] }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.7279em" }, children: _jsxs(_components.span, { style: { top: "-3.9029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1/" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mord", children: "∣" })] })] })] }) }), "\n", _jsx(_components.p, { children: "Therefore:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "|x| < 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", the series converges absolutely"] }), "\n", _jsxs(_components.li, { children: ["if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "|x| > 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", the series diverges"] }), "\n", _jsxs(_components.li, { children: ["if ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mi, { mathvariant: "normal", children: "∣" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "|x| = 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", the root test is inconclusive"] }), "\n"] }), "\n", _jsx(_components.p, { children: "Check the endpoints separately:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["at ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x=1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", we get ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsxs(_components.msup, { children: [_jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sum_{n=1}^{\\infty} (-1)^{n+1}\\frac{1}{n}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1901em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "+" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8451em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) })] }), ", which converges by the ", _jsx(_components.a, { href: "#leibniz-criterion", className: "internal", children: "#Leibniz Criterion" })] }), "\n", _jsxs(_components.li, { children: ["at ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "x=-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), ", we get ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsxs(_components.msubsup, { children: [_jsx(_components.mo, { children: "∑" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { children: "n" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "-\\sum_{n=1}^{\\infty} \\frac{1}{n}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1901em", verticalAlign: "-0.345em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, { className: "mop op-symbol small-op", style: { position: "relative", top: "0em" }, children: "∑" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8043em" }, children: [_jsxs(_components.span, { style: { top: "-2.4003em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "=" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }), _jsxs(_components.span, { style: { top: "-3.2029em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "∞" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2997em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8451em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) })] }), ", which diverges"] }), "\n"] }), "\n", _jsx(_components.p, { children: "So the interval of convergence is" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(-1,1]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }) }), "\n", _jsx(_components.span, { className: "katex-error", title: "ParseError: KaTeX parse error: Expected 'EOF', got '\\end' at position 70: …text{n is odd}\n\\̲e̲n̲d̲{cases}", style: { color: "#cc0000" }, children: "\\frac{1}{2^{n}} \\text{n is even} \\\\\n\\frac{-1}{3^{n}} \\text{n is odd}\n\\end{cases}" })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent };

Links to this page: