Leibniz-Reihe – Wikipedia

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width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>, die <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in den Jahren 1673–1676 entwickelte und 1682 in der Zeitschrift <i><a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a></i> erstmals veröffentlichte.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Sie lautet: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\dotsb ={\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\dotsb ={\frac {\pi }{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef36d20e884bc257c537b4610511fedac46f2e48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.402ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\dotsb ={\frac {\pi }{4}}}"></span>.</dd></dl> <p>Diese Formel war dem <a href="/wiki/Indische_Mathematik" class="mw-redirect" title="Indische Mathematik">indischen Mathematiker</a> <a href="/wiki/Madhava_(Mathematiker)" title="Madhava (Mathematiker)">Madhava</a> bereits im 14. Jahrhundert bekannt, weswegen sie mittlerweile auch <b>Madhava-Leibniz-Reihe</b> genannt wird, und dem schottischen Mathematiker <a href="/wiki/James_Gregory_(Mathematiker)" title="James Gregory (Mathematiker)">Gregory</a> bereits vor 1671; Leibniz entdeckte sie für die kontinentaleuropäische <a href="/wiki/Mathematik" title="Mathematik">Mathematik</a> neu. Die Reihe wird daher manchmal auch zusätzlich nach Gregory benannt. </p><p>Die <a href="/wiki/Konvergenz_(Mathematik)" title="Konvergenz (Mathematik)">Konvergenz</a> dieser <a href="/wiki/Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihe</a> folgt unmittelbar aus dem <a href="/wiki/Leibniz-Kriterium" title="Leibniz-Kriterium">Leibniz-Kriterium</a>. Die Konvergenz ist <a href="/wiki/Konvergenzgeschwindigkeit" title="Konvergenzgeschwindigkeit">logarithmisch</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Herleitung"><span class="tocnumber">1</span> <span class="toctext">Herleitung</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Konvergenzgeschwindigkeit"><span class="tocnumber">2</span> <span class="toctext">Konvergenzgeschwindigkeit</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Eine_Liste_von_Partialsummen,_die_sich_aus_Leibniz’_Formel_ergeben"><span class="tocnumber">3</span> <span class="toctext">Eine Liste von Partialsummen, die sich aus Leibniz’ Formel ergeben</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Konvergenz-Beschleunigung"><span class="tocnumber">4</span> <span class="toctext">Konvergenz-Beschleunigung</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Analoge_Abwandlungen"><span class="tocnumber">5</span> <span class="toctext">Analoge Abwandlungen</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Siehe_auch"><span class="tocnumber">6</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Literatur"><span class="tocnumber">7</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Weblinks"><span class="tocnumber">8</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Einzelnachweise"><span class="tocnumber">9</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=1" title="Abschnitt bearbeiten: Herleitung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Herleitung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Leibniz-Reihe kann so hergeleitet werden: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{2k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{2k}{\biggr ]}\mathrm {d} x=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{2k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{2k}{\biggr ]}\mathrm {d} x=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30a22373780159c248026f7f825bcc64fdff5363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.205ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{2k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{2k}{\biggr ]}\mathrm {d} x=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\int _{0}^{1}{\frac {1}{x^{2}+1}}\,\mathrm {d} x={\biggl [}\arctan(x){\biggr ]}_{x=0}^{x=1}=\arctan(1)={\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\int _{0}^{1}{\frac {1}{x^{2}+1}}\,\mathrm {d} x={\biggl [}\arctan(x){\biggr ]}_{x=0}^{x=1}=\arctan(1)={\frac {\pi }{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24813f10e9f5b6e2a6faf531b8c48335868946cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.213ex; height:6.509ex;" alt="{\displaystyle =\int _{0}^{1}{\frac {1}{x^{2}+1}}\,\mathrm {d} x={\biggl [}\arctan(x){\biggr ]}_{x=0}^{x=1}=\arctan(1)={\frac {\pi }{4}}}"></span></dd></dl> <p>Auf der <a href="/wiki/Geometrische_Reihe" title="Geometrische Reihe">Geometrischen Reihe</a> basiert die Umwandlung von der unendlichen Summe der Standard-Polynomfunktionen zur gezeigten gebrochen rationalen Funktion. </p> <div class="mw-heading mw-heading2"><h2 id="Konvergenzgeschwindigkeit">Konvergenzgeschwindigkeit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=2" title="Abschnitt bearbeiten: Konvergenzgeschwindigkeit" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Konvergenzgeschwindigkeit"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das Restglied der Summe nach <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> Summanden beträgt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}=\sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}-{\frac {\pi }{4}}=-\sum _{k=n}^{\infty }{\frac {(-1)^{k}}{2k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{n}=\sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}-{\frac {\pi }{4}}=-\sum _{k=n}^{\infty }{\frac {(-1)^{k}}{2k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6ec4014ccf9a666a95ff9a3edd4b97ec7bf35b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.293ex; height:7.509ex;" alt="{\displaystyle R_{n}=\sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}-{\frac {\pi }{4}}=-\sum _{k=n}^{\infty }{\frac {(-1)^{k}}{2k+1}}}"></span>.</dd></dl> <p>Mit der Fehlerabschätzung des <a href="/wiki/Leibniz-Kriterium" title="Leibniz-Kriterium">Leibniz-Kriteriums</a> gilt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |R_{n}|\leq {\frac {1}{2n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{n}|\leq {\frac {1}{2n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f26fb5f99a259f7da745633a05b04743fa5d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.771ex; height:5.343ex;" alt="{\displaystyle |R_{n}|\leq {\frac {1}{2n+1}}}"></span>.</dd></dl> <p>Genauere Betrachtungen zeigen sogar, dass </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |R_{n}|<{\frac {1}{4n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R_{n}|<{\frac {1}{4n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276a7a4e21c3e70e345ac88379091548cd6b152d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.768ex; height:5.176ex;" alt="{\displaystyle |R_{n}|<{\frac {1}{4n}}}"></span>.</dd></dl> <p>Mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> Summanden kann man also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> Nachkommastellen mit einem Fehler < 0,5 in der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>-ten Nachkommastelle erhalten: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(n)=\lg(2n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>lg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n)=\lg(2n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a346c9b41e2eb09a6d852f6a62b7bed11a6d197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.569ex; height:2.843ex;" alt="{\displaystyle s(n)=\lg(2n)}"></span>.</dd></dl> <p>Die Anzahl benötigter Summanden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> sinnvolle Nachkommastellen im Ergebnis beträgt entsprechend </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(s)={\frac {1}{2}}\cdot 10^{\textstyle s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(s)={\frac {1}{2}}\cdot 10^{\textstyle s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85cea9ed208b32c8f609899301d3ffbbad06cd0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.718ex; height:5.176ex;" alt="{\displaystyle n(s)={\frac {1}{2}}\cdot 10^{\textstyle s}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Eine_Liste_von_Partialsummen,_die_sich_aus_Leibniz’_Formel_ergeben"><span id="Eine_Liste_von_Partialsummen.2C_die_sich_aus_Leibniz.E2.80.99_Formel_ergeben"></span>Eine Liste von Partialsummen, die sich aus Leibniz’ Formel ergeben</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=3" title="Abschnitt bearbeiten: Eine Liste von Partialsummen, die sich aus Leibniz’ Formel ergeben" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Eine Liste von Partialsummen, die sich aus Leibniz’ Formel ergeben"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mit Hilfe der Leibniz-Reihe lässt sich eine Näherung der Kreiszahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> berechnen, denn es ist </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =4\cdot \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=\lim \limits _{n\to \infty }\left(4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =4\cdot \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=\lim \limits _{n\to \infty }\left(4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a2b029a9a24538ff65498e61ad50a794ee486d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.462ex; height:7.509ex;" alt="{\displaystyle \pi =4\cdot \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=\lim \limits _{n\to \infty }\left(4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}\right)}"></span>.</dd></dl> <p>Die folgende Liste zeigt die Folgenglieder der Folge von <a href="/wiki/Partialsumme" class="mw-redirect" title="Partialsumme">Partialsummen</a> der mit 4 multiplizierten Leibniz-Reihe. </p><p>Da die Folge nur sehr langsam konvergiert, ist sie zur effizienten Berechnung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> nicht geeignet, auch nicht nach Umformungen. </p><p>Bemerkenswert ist die Tatsache, dass in der letzten Tabellenzeile die 9. Nachkommastelle noch nicht richtig ist, hingegen die nächsten 6 (...589793...) mit der Kreiszahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> übereinstimmen. </p> <dl><dd><table class="wikitable" style="text-align:right;"> <tbody><tr> <th>n<br />(Anzahl der<br />berechneten<br />Brüche) </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348e287d95b58299d8ada48adcf9bceec800a445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.796ex; height:7.509ex;" alt="{\displaystyle 4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}}"></span><br /><br />(Ergebnis) </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot \sum _{k=0}^{n}{\frac {(-1)^{k}}{2k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot \sum _{k=0}^{n}{\frac {(-1)^{k}}{2k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0186d79053161e4e3348a066681e4af22ac71bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.796ex; height:7.176ex;" alt="{\displaystyle 4\cdot \sum _{k=0}^{n}{\frac {(-1)^{k}}{2k+1}}}"></span><br /><br />(Ergebnis) </th> <th>Mittelwert </th></tr> <tr> <td>2</td> <td><span style="color:#AA0000;">2,6666666666666665</span></td> <td>3,<span style="color:#AA0000;">4666666666666668</span></td> <td>3,<span style="color:#AA0000;">0666666666666664</span> </td></tr> <tr> <td>4</td> <td><span style="color:#AA0000;">2,8952380952380952</span></td> <td>3,<span style="color:#AA0000;">3396825396825394</span></td> <td>3,1<span style="color:#AA0000;">174603174603175</span> </td></tr> <tr> <td>8</td> <td>3,<span style="color:#AA0000;">0170718170718169</span></td> <td>3,<span style="color:#AA0000;">2523659347188758</span></td> <td>3,1<span style="color:#AA0000;">347188758953464</span> </td></tr> <tr> <td>16</td> <td>3,<span style="color:#AA0000;">0791533941974261</span></td> <td>3,<span style="color:#AA0000;">2003655154095472</span></td> <td>3,1<span style="color:#AA0000;">397594548034866</span> </td></tr> <tr> <td>32</td> <td>3,1<span style="color:#AA0000;">103502736986859</span></td> <td>3,1<span style="color:#AA0000;">718887352371476</span></td> <td>3,141<span style="color:#AA0000;">1195044679165</span> </td></tr> <tr> <td>64</td> <td>3,1<span style="color:#AA0000;">259686069732875</span></td> <td>3,1<span style="color:#AA0000;">569763589112720</span></td> <td>3,141<span style="color:#AA0000;">4724829422798</span> </td></tr> <tr> <td>100</td> <td>3,1<span style="color:#AA0000;">315929035585528</span></td> <td>3,1<span style="color:#AA0000;">514934010709905</span></td> <td>3,1415<span style="color:#AA0000;">431523147719</span> </td></tr> <tr> <td>1000</td> <td>3,14<span style="color:#AA0000;">05926538397928</span></td> <td>3,14<span style="color:#AA0000;">25916543395429</span></td> <td>3,141592<span style="color:#AA0000;">1540896679</span> </td></tr> <tr> <td>10000</td> <td>3,141<span style="color:#AA0000;">4926535900429</span></td> <td>3,141<span style="color:#AA0000;">6926435905430</span></td> <td>3,1415926<span style="color:#AA0000;">485902927</span> </td></tr> <tr> <td>100000</td> <td>3,1415<span style="color:#AA0000;">826535897935</span></td> <td>3,141<span style="color:#AA0000;">6026534897941</span></td> <td>3,1415926535<span style="color:#AA0000;">397936</span> </td></tr> <tr> <td>1000000</td> <td>3,14159<span style="color:#AA0000;">16535897930</span></td> <td>3,14159<span style="color:#AA0000;">36535887932</span></td> <td>3,141592653589<span style="color:#AA0000;">2931</span> </td></tr> <tr> <td>10000000</td> <td>3,141592<span style="color:#AA0000;">5535897928</span></td> <td>3,141592<span style="color:#AA0000;">7535897827</span></td> <td>3,1415926535897<span style="color:#AA0000;">878</span> </td></tr> <tr> <td>100000000</td> <td>3,1415926<span style="color:#AA0000;">435897932</span></td> <td>3,1415926<span style="color:#AA0000;">635897931</span></td> <td>3,141592653589793<span style="color:#AA0000;">1</span> </td></tr> <tr> <td>1000000000</td> <td>3,14159265<span style="color:#AA0000;">25897930</span></td> <td>3,14159265<span style="color:#AA0000;">45897932</span></td> <td>3,141592653589793<span style="color:#AA0000;">1</span> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Konvergenz-Beschleunigung">Konvergenz-Beschleunigung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=4" title="Abschnitt bearbeiten: Konvergenz-Beschleunigung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Konvergenz-Beschleunigung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die <a href="/wiki/Eulersche_Reihentransformation" title="Eulersche Reihentransformation">Eulersche Reihentransformation</a> erzeugt aus der Leibniz-Reihe die schneller konvergierende Reihe (<a href="/wiki/Nicolas_Fatio_de_Duillier" title="Nicolas Fatio de Duillier">Nicolas Fatio</a>, 1705) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}={\frac {1}{2}}\left(1+{\frac {1}{1\cdot 3}}+{\frac {1\cdot 2}{1\cdot 3\cdot 5}}+...+{\frac {1\cdot 2...n}{1\cdot 3\cdot 5...(2n+1)}}+...\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> </mrow> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2...</mn> <mi>n</mi> </mrow> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5...</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}={\frac {1}{2}}\left(1+{\frac {1}{1\cdot 3}}+{\frac {1\cdot 2}{1\cdot 3\cdot 5}}+...+{\frac {1\cdot 2...n}{1\cdot 3\cdot 5...(2n+1)}}+...\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9f54c8e8e6625c1dc8e66e81d2cc179a2a5aba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:61.092ex; height:6.343ex;" alt="{\displaystyle {\frac {\pi }{4}}={\frac {1}{2}}\left(1+{\frac {1}{1\cdot 3}}+{\frac {1\cdot 2}{1\cdot 3\cdot 5}}+...+{\frac {1\cdot 2...n}{1\cdot 3\cdot 5...(2n+1)}}+...\right).}"></span></dd></dl> <p>Verbesserte Verfahren mit anderen Reihen sind im Artikel <a href="/wiki/Kreiszahl" title="Kreiszahl">Kreiszahl</a> aufgeführt. </p> <div class="mw-heading mw-heading2"><h2 id="Analoge_Abwandlungen">Analoge Abwandlungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=5" title="Abschnitt bearbeiten: Analoge Abwandlungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Analoge Abwandlungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zur Leibniz-Reihe können einige analoge Abwandlungen erstellt werden.<br /> Das bekannteste Analogon ist die unendliche alternierende Differenz aller natürlicher Zahlen, welche direkt zum Logarithmus Naturalis von Zwei führt: </p> <dl><dd><table class="wikitable"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}\approx 0{,}6931471805599453=\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0,693</mn> <mn>1471805599453</mn> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}\approx 0{,}6931471805599453=\ln(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d156a8a41347b89be5eac32b1989a37b76ec1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:66.305ex; height:7.176ex;" alt="{\displaystyle 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}\approx 0{,}6931471805599453=\ln(2)}"></span> </td></tr></tbody></table></dd></dl> <p>Herleitung dieses Wertes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x+1}}\,\mathrm {d} x=\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x+1}}\,\mathrm {d} x=\ln(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09469adad2875e9347cb9a799358923bd8bc63f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:80.791ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x+1}}\,\mathrm {d} x=\ln(2)}"></span><br /></dd></dl> <p>Dies ist ein kubisches Analogon zur Leibniz-Reihe: </p> <dl><dd><table class="wikitable"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\frac {1}{4}}+{\frac {1}{7}}-{\frac {1}{10}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{3k+1}}\approx 0{,}835648848264721={\frac {1}{9}}{\sqrt {3}}\pi +{\frac {1}{3}}\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0,835</mn> <mn>648848264721</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>π</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\frac {1}{4}}+{\frac {1}{7}}-{\frac {1}{10}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{3k+1}}\approx 0{,}835648848264721={\frac {1}{9}}{\sqrt {3}}\pi +{\frac {1}{3}}\ln(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e582690b03b7c70bc64bf5924523dd4aee35945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.468ex; height:7.176ex;" alt="{\displaystyle 1-{\frac {1}{4}}+{\frac {1}{7}}-{\frac {1}{10}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{3k+1}}\approx 0{,}835648848264721={\frac {1}{9}}{\sqrt {3}}\pi +{\frac {1}{3}}\ln(2)}"></span> </td></tr></tbody></table></dd></dl> <p>Herleitung dieses Wertes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{3k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{3k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{3k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x^{3}+1}}\,\mathrm {d} x=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{3k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{3k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{3k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x^{3}+1}}\,\mathrm {d} x=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a863b85697e7dcd0db2965b0be7215b2d0ecc16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.441ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{3k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{3k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{3k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x^{3}+1}}\,\mathrm {d} x=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\int _{0}^{1}{\frac {1}{2}}\,{\frac {1}{x^{2}-x+1}}+{\frac {1}{3}}\,{\frac {1}{x+1}}-{\frac {1}{6}}\,{\frac {2x-1}{x^{2}-x+1}}\,\mathrm {d} x=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\int _{0}^{1}{\frac {1}{2}}\,{\frac {1}{x^{2}-x+1}}+{\frac {1}{3}}\,{\frac {1}{x+1}}-{\frac {1}{6}}\,{\frac {2x-1}{x^{2}-x+1}}\,\mathrm {d} x=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da40257b8607dd79da849ecb3518e656fcceb4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.614ex; height:6.176ex;" alt="{\displaystyle =\int _{0}^{1}{\frac {1}{2}}\,{\frac {1}{x^{2}-x+1}}+{\frac {1}{3}}\,{\frac {1}{x+1}}-{\frac {1}{6}}\,{\frac {2x-1}{x^{2}-x+1}}\,\mathrm {d} x=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\biggl \{}{\frac {1}{3}}{\sqrt {3}}\arctan {\bigl [}{\frac {1}{3}}{\sqrt {3}}\,(2x-1){\bigr ]}+{\frac {1}{6}}\ln {\biggl [}{\frac {(x+1)^{2}}{x^{2}-x+1}}{\biggr ]}{\biggr \}}_{x=0}^{x=1}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\biggl \{}{\frac {1}{3}}{\sqrt {3}}\arctan {\bigl [}{\frac {1}{3}}{\sqrt {3}}\,(2x-1){\bigr ]}+{\frac {1}{6}}\ln {\biggl [}{\frac {(x+1)^{2}}{x^{2}-x+1}}{\biggr ]}{\biggr \}}_{x=0}^{x=1}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26e0d13ec27a719530baf4992c12304ac6eb0da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.131ex; height:6.509ex;" alt="{\displaystyle ={\biggl \{}{\frac {1}{3}}{\sqrt {3}}\arctan {\bigl [}{\frac {1}{3}}{\sqrt {3}}\,(2x-1){\bigr ]}+{\frac {1}{6}}\ln {\biggl [}{\frac {(x+1)^{2}}{x^{2}-x+1}}{\biggr ]}{\biggr \}}_{x=0}^{x=1}=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{3}}{\sqrt {3}}\arctan {\bigl (}{\frac {1}{3}}{\sqrt {3}}{\bigr )}-{\frac {1}{3}}{\sqrt {3}}\arctan {\bigl (}-{\frac {1}{3}}{\sqrt {3}}{\bigr )}+{\frac {1}{6}}\ln(4)={\frac {1}{9}}{\sqrt {3}}\,\pi +{\frac {1}{3}}\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{3}}{\sqrt {3}}\arctan {\bigl (}{\frac {1}{3}}{\sqrt {3}}{\bigr )}-{\frac {1}{3}}{\sqrt {3}}\arctan {\bigl (}-{\frac {1}{3}}{\sqrt {3}}{\bigr )}+{\frac {1}{6}}\ln(4)={\frac {1}{9}}{\sqrt {3}}\,\pi +{\frac {1}{3}}\ln(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f891b8dde8b5065850dd91459ea3245b24dd1789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:77.451ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{3}}{\sqrt {3}}\arctan {\bigl (}{\frac {1}{3}}{\sqrt {3}}{\bigr )}-{\frac {1}{3}}{\sqrt {3}}\arctan {\bigl (}-{\frac {1}{3}}{\sqrt {3}}{\bigr )}+{\frac {1}{6}}\ln(4)={\frac {1}{9}}{\sqrt {3}}\,\pi +{\frac {1}{3}}\ln(2)}"></span><br /></dd></dl> <p>Und das ist ein quartisches Analogon zur Leibniz-Reihe: </p> <dl><dd><table class="wikitable"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-{\frac {1}{5}}+{\frac {1}{9}}-{\frac {1}{13}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4k+1}}\approx 0{,}866972987339911={\frac {1}{8}}{\sqrt {2}}\,\pi +{\frac {1}{4}}{\sqrt {2}}\operatorname {arsinh} (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>13</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0,866</mn> <mn>972987339911</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>arsinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-{\frac {1}{5}}+{\frac {1}{9}}-{\frac {1}{13}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4k+1}}\approx 0{,}866972987339911={\frac {1}{8}}{\sqrt {2}}\,\pi +{\frac {1}{4}}{\sqrt {2}}\operatorname {arsinh} (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefbf31e2494124ac85a83c38ab56101bfeee7a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:86.236ex; height:7.176ex;" alt="{\displaystyle 1-{\frac {1}{5}}+{\frac {1}{9}}-{\frac {1}{13}}+\dotsb =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4k+1}}\approx 0{,}866972987339911={\frac {1}{8}}{\sqrt {2}}\,\pi +{\frac {1}{4}}{\sqrt {2}}\operatorname {arsinh} (1)}"></span> </td></tr></tbody></table></dd></dl> <p>Herleitung dieses Wertes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{4k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{4k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x^{4}+1}}\,\mathrm {d} x=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{4k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{4k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x^{4}+1}}\,\mathrm {d} x=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe1df68e5afee63035ad3f46a9eaa04c329ce3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.441ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4k+1}}=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{1}x^{4k}\,\mathrm {d} x=\int _{0}^{1}{\biggl [}\sum _{k=0}^{\infty }(-1)^{k}x^{4k}{\biggr ]}\mathrm {d} x=\int _{0}^{1}{\frac {1}{x^{4}+1}}\,\mathrm {d} x=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\int _{0}^{1}{\frac {1}{4}}\,{\frac {1}{x^{2}+{\sqrt {2}}\,x+1}}+{\frac {1}{4}}\,{\frac {1}{x^{2}-{\sqrt {2}}\,x+1}}+{\frac {1}{8}}{\sqrt {2}}\,{\frac {2x+{\sqrt {2}}}{x^{2}+{\sqrt {2}}\,x+1}}-{\frac {1}{8}}{\sqrt {2}}\,{\frac {2x-{\sqrt {2}}}{x^{2}-{\sqrt {2}}\,x+1}}\,\mathrm {d} x=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\int _{0}^{1}{\frac {1}{4}}\,{\frac {1}{x^{2}+{\sqrt {2}}\,x+1}}+{\frac {1}{4}}\,{\frac {1}{x^{2}-{\sqrt {2}}\,x+1}}+{\frac {1}{8}}{\sqrt {2}}\,{\frac {2x+{\sqrt {2}}}{x^{2}+{\sqrt {2}}\,x+1}}-{\frac {1}{8}}{\sqrt {2}}\,{\frac {2x-{\sqrt {2}}}{x^{2}-{\sqrt {2}}\,x+1}}\,\mathrm {d} x=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/673663b0e2982081c887a98da5439a49cdce3b1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:95.596ex; height:6.843ex;" alt="{\displaystyle =\int _{0}^{1}{\frac {1}{4}}\,{\frac {1}{x^{2}+{\sqrt {2}}\,x+1}}+{\frac {1}{4}}\,{\frac {1}{x^{2}-{\sqrt {2}}\,x+1}}+{\frac {1}{8}}{\sqrt {2}}\,{\frac {2x+{\sqrt {2}}}{x^{2}+{\sqrt {2}}\,x+1}}-{\frac {1}{8}}{\sqrt {2}}\,{\frac {2x-{\sqrt {2}}}{x^{2}-{\sqrt {2}}\,x+1}}\,\mathrm {d} x=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\biggl [}{\frac {1}{2}}{\sqrt {2}}\arctan {\bigl (}{\frac {{\sqrt {2}}\,x}{{\sqrt {x^{4}+1}}-x^{2}+1}}{\bigr )}+{\frac {1}{8}}{\sqrt {2}}\ln {\bigl (}{\frac {x^{2}+{\sqrt {2}}\,x+1}{x^{2}-{\sqrt {2}}\,x+1}}{\bigr )}{\biggr ]}_{x=0}^{x=1}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\biggl [}{\frac {1}{2}}{\sqrt {2}}\arctan {\bigl (}{\frac {{\sqrt {2}}\,x}{{\sqrt {x^{4}+1}}-x^{2}+1}}{\bigr )}+{\frac {1}{8}}{\sqrt {2}}\ln {\bigl (}{\frac {x^{2}+{\sqrt {2}}\,x+1}{x^{2}-{\sqrt {2}}\,x+1}}{\bigr )}{\biggr ]}_{x=0}^{x=1}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03fe17cc4c45d001d9a1098af134a4a4a5a0b232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.535ex; height:7.176ex;" alt="{\displaystyle ={\biggl [}{\frac {1}{2}}{\sqrt {2}}\arctan {\bigl (}{\frac {{\sqrt {2}}\,x}{{\sqrt {x^{4}+1}}-x^{2}+1}}{\bigr )}+{\frac {1}{8}}{\sqrt {2}}\ln {\bigl (}{\frac {x^{2}+{\sqrt {2}}\,x+1}{x^{2}-{\sqrt {2}}\,x+1}}{\bigr )}{\biggr ]}_{x=0}^{x=1}=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{2}}{\sqrt {2}}\arctan(1)+{\frac {1}{8}}{\sqrt {2}}\ln {\bigl (}{\frac {2+{\sqrt {2}}}{2-{\sqrt {2}}}}{\bigr )}={\frac {1}{8}}{\sqrt {2}}\,\pi +{\frac {1}{4}}{\sqrt {2}}\operatorname {arsinh} (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>arsinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}{\sqrt {2}}\arctan(1)+{\frac {1}{8}}{\sqrt {2}}\ln {\bigl (}{\frac {2+{\sqrt {2}}}{2-{\sqrt {2}}}}{\bigr )}={\frac {1}{8}}{\sqrt {2}}\,\pi +{\frac {1}{4}}{\sqrt {2}}\operatorname {arsinh} (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbde9ef6168ee30460335c79cfcbe1d08fc9bf99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:65.526ex; height:6.843ex;" alt="{\displaystyle ={\frac {1}{2}}{\sqrt {2}}\arctan(1)+{\frac {1}{8}}{\sqrt {2}}\ln {\bigl (}{\frac {2+{\sqrt {2}}}{2-{\sqrt {2}}}}{\bigr )}={\frac {1}{8}}{\sqrt {2}}\,\pi +{\frac {1}{4}}{\sqrt {2}}\operatorname {arsinh} (1)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=6" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wallissches_Produkt" title="Wallissches Produkt">Wallissches Produkt</a></li> <li><a href="/wiki/Arkustangens_und_Arkuskotangens#Reihenentwicklungen" title="Arkustangens und Arkuskotangens">Arkustangens</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=7" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>V. I. Bityutskov: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Leibniz series</cite>. In: <a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="/wiki/Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, <a href="/wiki/Spezial:ISBN-Suche/1556080107" class="internal mw-magiclink-isbn">ISBN 1-55608-010-7</a> (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/leibniz_series">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Leibniz-Reihe&rft.atitle=Leibniz+series&rft.au=V.+I.+Bityutskov&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=8" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>3Blue1Brown: <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=NaL_Cb42WyY&t"><i>Pi hiding in prime regularities</i></a> auf <a href="/wiki/YouTube" title="YouTube">YouTube</a>, abgerufen am 18. Oktober 2023 (<a href="/wiki/Zahlentheorie" title="Zahlentheorie">Zahlentheoretischer</a> Zugang zur Leibniz-Reihe).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Leibniz-Reihe&veaction=edit&section=9" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Leibniz-Reihe&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Gottfried Wilhelm Freiherr von Leibniz: <cite style="font-style:italic">Leibnizens mathematische Schriften: Mathematik</cite>. A. Asher, 1858 (<a rel="nofollow" class="external text" href="https://books.google.it/books?id=WNUNAQAAIAAJ&pg=PA118">google.it</a> [abgerufen am 31. Januar 2023]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Leibniz-Reihe&rft.au=Gottfried+Wilhelm+Freiherr+von+Leibniz&rft.btitle=Leibnizens+mathematische+Schriften%3A+Mathematik&rft.date=1858&rft.genre=book&rft.pub=A.+Asher" style="display:none"> </span></span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Leibniz-Reihe&oldid=245837680">https://de.wikipedia.org/w/index.php?title=Leibniz-Reihe&oldid=245837680</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Folgen_und_Reihen" title="Kategorie:Folgen und Reihen">Folgen und Reihen</a></li><li><a href="/wiki/Kategorie:Gottfried_Wilhelm_Leibniz_als_Namensgeber" title="Kategorie:Gottfried Wilhelm Leibniz als Namensgeber">Gottfried Wilhelm Leibniz als Namensgeber</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Leibniz-Reihe" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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hreflang="ca" data-title="Sèrie de Leibniz" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Leibniz%27_r%C3%A6kke" title="Leibniz' række – Dänisch" lang="da" hreflang="da" data-title="Leibniz' række" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80" title="Leibniz formula for π – Englisch" lang="en" hreflang="en" data-title="Leibniz formula for π" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Serie_de_Leibniz" title="Serie de Leibniz – Spanisch" lang="es" hreflang="es" data-title="Serie de Leibniz" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Leibnizen_serie" title="Leibnizen serie – Baskisch" lang="eu" hreflang="eu" data-title="Leibnizen serie" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%95%D7%A1%D7%97%D7%AA_%D7%9C%D7%99%D7%99%D7%91%D7%A0%D7%99%D7%A5_%D7%9C-%CF%80" title="נוסחת לייבניץ ל-π – Hebräisch" lang="he" hreflang="he" data-title="נוסחת לייבניץ ל-π" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BE%E0%A4%88_%E0%A4%95%E0%A5%87_%E0%A4%B2%E0%A4%BF%E0%A4%AF%E0%A5%87_%E0%A4%AE%E0%A4%BE%E0%A4%A7%E0%A4%B5-%E0%A4%B2%E0%A5%88%E0%A4%AC%E0%A4%A8%E0%A5%80%E0%A4%9C_%E0%A4%B8%E0%A5%82%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="पाई के लिये माधव-लैबनीज सूत्र – Hindi" lang="hi" hreflang="hi" data-title="पाई के लिये माधव-लैबनीज सूत्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BC%D5%A1%D5%B5%D5%A2%D5%B6%D5%AB%D6%81%D5%AB_%D5%B7%D5%A1%D6%80%D6%84" title="Լայբնիցի շարք – Armenisch" lang="hy" hreflang="hy" data-title="Լայբնիցի շարք" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Formula_di_Leibniz_per_pi" title="Formula di Leibniz per pi – Italienisch" lang="it" hreflang="it" data-title="Formula di Leibniz per pi" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%82%A4%E3%83%97%E3%83%8B%E3%83%83%E3%83%84%E3%81%AE%E5%85%AC%E5%BC%8F" title="ライプニッツの公式 – Japanisch" lang="ja" hreflang="ja" data-title="ライプニッツの公式" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9D%BC%EC%9D%B4%ED%94%84%EB%8B%88%EC%B8%A0%EC%9D%98_%EC%9B%90%EC%A3%BC%EC%9C%A8_%EA%B3%B5%EC%8B%9D" title="라이프니츠의 원주율 공식 – Koreanisch" lang="ko" hreflang="ko" data-title="라이프니츠의 원주율 공식" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/F%C3%B3rmula_de_Leibniz_para_%CF%80" title="Fórmula de Leibniz para π – Portugiesisch" lang="pt" hreflang="pt" data-title="Fórmula de Leibniz para π" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Formula_lui_Leibniz_pentru_%CF%80" title="Formula lui Leibniz pentru π – Rumänisch" lang="ro" hreflang="ro" data-title="Formula lui Leibniz pentru π" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D1%8F%D0%B4_%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D0%B8%D1%86%D0%B0" title="Ряд Лейбница – Russisch" lang="ru" hreflang="ru" data-title="Ряд Лейбница" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Formula_e_Leibniz-it_p%C3%ABr_%CF%80" title="Formula e Leibniz-it për π – Albanisch" lang="sq" hreflang="sq" data-title="Formula e Leibniz-it për π" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%8F%D0%B4_%D0%9B%D0%B5%D0%B9%D0%B1%D0%BD%D1%96%D1%86%D0%B0" title="Ряд Лейбніца – Ukrainisch" lang="uk" hreflang="uk" data-title="Ряд Лейбніца" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%B4ng_th%E1%BB%A9c_Leibniz_%C4%91%E1%BB%83_t%C3%ADnh_%CF%80" title="Công thức Leibniz để tính π – Vietnamesisch" lang="vi" hreflang="vi" data-title="Công thức Leibniz để tính π" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%CE%A0%E7%9A%84%E8%8E%B1%E5%B8%83%E5%B0%BC%E8%8C%A8%E5%85%AC%E5%BC%8F" title="Π的莱布尼茨公式 – Chinesisch" lang="zh" hreflang="zh" data-title="Π的莱布尼茨公式" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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Leibniz-Reihe – Wikipedia