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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Differenzenrechnung</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Differenzenrechnung</b> ist ein Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>, das die <a href="Diskret" title="Diskret">diskrete</a> Entsprechung zur <a href="Analysis" title="Analysis">Analysis</a> (<a href="Differentialrechnung" title="Differentialrechnung">Differenzial-</a> und <a href="Integralrechnung" title="Integralrechnung">Integralrechnung</a>) bildet. Während sich die Analysis mit Funktionen beschäftigt, die auf kontinuierlichen Räumen definiert sind (um einen Grenzwertbegriff etablieren zu können), im Besonderen mit Funktionen auf den <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a>, interessiert man sich in der Differenzenrechnung für Funktionen auf den <a href="Ganze_Zahl" title="Ganze Zahl">ganzen Zahlen</a> ℤ. Die Differenzenrechnung kann zur Berechnung von <a href="Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihen</a> angewandt werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Differenzen_und_Summen">Differenzen und Summen</h2></div>
<p>Die bekannte <a href="Kontinuum_(Mathematik)" title="Kontinuum (Mathematik)">kontinuierliche</a> Differentialrechnung basiert auf dem <a href="Differential_(Mathematik)" title="Differential (Mathematik)">Differenzialoperator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {D} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c894024162d89aa8502758b2151cb579806ea9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathrm {D} }" loading="lazy"></span>, der wie folgt definiert ist:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} f(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">D</mi>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>h</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>h</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {D} f(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f350247c9faa30ba918d8412dd14d170ac254f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.782ex; height:5.843ex;" alt="{\displaystyle \mathrm {D} f(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}" loading="lazy"></span></dd></dl>
<p>Die Differenzenrechnung hingegen verwendet einen sogenannten <i>Differenzenoperator</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f(x)=f(x+1)-f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta f(x)=f(x+1)-f(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f476f29e065f1a1e0f67e13e89fb2a8f07cad235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.13ex; height:2.843ex;" alt="{\displaystyle \Delta f(x)=f(x+1)-f(x)}" loading="lazy"></span>.</dd></dl>
<p>Die umgekehrte Operation wird nicht wie in der kontinuierlichen Differentialrechnung mit dem <a href="Integralrechnung" title="Integralrechnung">unbestimmten Integral</a>, sondern mit einer <i>unbestimmten Summe</i> <span class="mwe-math-element mwe-math-element-inline"><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 f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∑<!-- ∑ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum f(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4db83f33004a4f57a476db8ceee5521180705341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.16ex; height:3.843ex;" alt="{\displaystyle \sum f(x)}" loading="lazy"></span> erreicht, die sich zum Differenzenoperator wie folgt verhält:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\Delta f(x)\quad \Longleftrightarrow \quad \sum g(x)\;\delta x=f(x)+C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="1em"></mspace>
<mo stretchy="false">⟺<!-- ⟺ --></mo>
<mspace width="1em"></mspace>
<mo>∑<!-- ∑ --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<mi>δ<!-- δ --></mi>
<mi>x</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(x)=\Delta f(x)\quad \Longleftrightarrow \quad \sum g(x)\;\delta x=f(x)+C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e479f1f387c725a31b97c66ad0a341777912623.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:47.102ex; height:3.843ex;" alt="{\displaystyle g(x)=\Delta f(x)\quad \Longleftrightarrow \quad \sum g(x)\;\delta x=f(x)+C}" loading="lazy"></span>.</dd></dl>
<p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span> verhält sich hier zu <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span> wie <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a15022657616b297a2c995d1b314a3aa3442c0cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} }" loading="lazy"></span> zu <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {D} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c894024162d89aa8502758b2151cb579806ea9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathrm {D} }" loading="lazy"></span> in der kontinuierlichen Differentialrechnung. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> steht für den Wert einer beliebigen Funktion, die für ganzzahlige <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> konstant ist (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x+1)=C(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C(x+1)=C(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37438a95c3b227fc401a410b3c9fffe060820dc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.912ex; height:2.843ex;" alt="{\displaystyle C(x+1)=C(x)}" loading="lazy"></span>).
</p><p>Das Pendant zu bestimmten Integralen sind <i>bestimmte Summen.</i> Diese entsprechen gewöhnlichen Summen ohne den Wert am höchsten Index:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {}{_{a}^{b}}\sum f(x)\;\delta x=\sum _{k=a}^{b-1}f(k)=[F(x)]_{a}^{b}=F(b)-F(a)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-OP">
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mrow class="MJX-TeXAtom-ORD">
<mpadded width="0">
<mrow class="MJX-TeXAtom-ORD">
<mphantom>
<mo>∑<!-- ∑ --></mo>
</mphantom>
</mrow>
</mpadded>
</mrow>
</mrow>
<mspace width="negativethinmathspace"></mspace>
<msubsup>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo movablelimits="false">∑<!-- ∑ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msubsup>
</mrow>
<mo><!-- --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<mi>δ<!-- δ --></mi>
<mi>x</mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</munderover>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msubsup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msubsup>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sideset {}{_{a}^{b}}\sum f(x)\;\delta x=\sum _{k=a}^{b-1}f(k)=[F(x)]_{a}^{b}=F(b)-F(a)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0ea82765e9a6615d61a752713491f365cd87a3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.387ex; width:49.452ex; height:7.343ex;" alt="{\displaystyle \sideset {}{_{a}^{b}}\sum f(x)\;\delta x=\sum _{k=a}^{b-1}f(k)=[F(x)]_{a}^{b}=F(b)-F(a)}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Invariante_Funktion">Invariante Funktion</h3></div>
<p>Eine unter dem Differenzialoperator <a href="Invariante_(Mathematik)" title="Invariante (Mathematik)">invariante</a> Funktion ist die <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> der Basis <i><a href="Eulersche_Zahl" title="Eulersche Zahl">e</a>.</i> In der Differenzenrechnung ist die Exponentialfunktion der Basis 2 invariant, wie sich leicht ermitteln lässt:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\Delta f(x)=f(x)\\\Longleftrightarrow \quad &f(x+1)-f(x)=f(x)\\\Longleftrightarrow \quad &f(x+1)=2f(x)\\\Longleftrightarrow \quad &\exists C:f(x)=C\cdot 2^{x}\\\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd></mtd>
<mtd>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⟺<!-- ⟺ --></mo>
<mspace width="1em"></mspace>
</mtd>
<mtd>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⟺<!-- ⟺ --></mo>
<mspace width="1em"></mspace>
</mtd>
<mtd>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⟺<!-- ⟺ --></mo>
<mspace width="1em"></mspace>
</mtd>
<mtd>
<mi mathvariant="normal">∃<!-- ∃ --></mi>
<mi>C</mi>
<mo>:</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>C</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msup>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\Delta f(x)=f(x)\\\Longleftrightarrow \quad &f(x+1)-f(x)=f(x)\\\Longleftrightarrow \quad &f(x+1)=2f(x)\\\Longleftrightarrow \quad &\exists C:f(x)=C\cdot 2^{x}\\\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a40959061991349375d49732f314b939d8bcac9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:30.585ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}&\Delta f(x)=f(x)\\\Longleftrightarrow \quad &f(x+1)-f(x)=f(x)\\\Longleftrightarrow \quad &f(x+1)=2f(x)\\\Longleftrightarrow \quad &\exists C:f(x)=C\cdot 2^{x}\\\end{aligned}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Fallende_Fakultäten"><span id="Fallende_Fakult.C3.A4ten"></span>Fallende Fakultäten</h3></div>
<p>Eine einfache Rechenregel gibt es für <a href="Pochhammer-Symbol" title="Pochhammer-Symbol">fallende Fakultäten</a>, die für jede Ganzzahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> wie folgt definiert sind:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\underline {m}}={\frac {x!}{(x-m)!}}={\begin{cases}\overbrace {x(x-1)\ldots (x-m+1)} ^{m{\text{ Faktoren}}}&{\text{, wenn }}m\geq 0\\&\\\underbrace {\frac {1}{(x+1)(x+2)\ldots (x-m)}} _{|m|{\text{ Faktoren}}}&{\text{, wenn }}m<0\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mi>m</mi>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mover>
<mrow>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>m</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏞<!-- ⏞ --></mo>
</mover>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext> Faktoren</mtext>
</mrow>
</mrow>
</mover>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>, wenn </mtext>
</mrow>
<mi>m</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext> Faktoren</mtext>
</mrow>
</mrow>
</munder>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>, wenn </mtext>
</mrow>
<mi>m</mi>
<mo><</mo>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{\underline {m}}={\frac {x!}{(x-m)!}}={\begin{cases}\overbrace {x(x-1)\ldots (x-m+1)} ^{m{\text{ Faktoren}}}&{\text{, wenn }}m\geq 0\\&\\\underbrace {\frac {1}{(x+1)(x+2)\ldots (x-m)}} _{|m|{\text{ Faktoren}}}&{\text{, wenn }}m<0\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e622a343ce6ca4424ac6a379ad08c893362cd8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:61.243ex; height:17.176ex;" alt="{\displaystyle x^{\underline {m}}={\frac {x!}{(x-m)!}}={\begin{cases}\overbrace {x(x-1)\ldots (x-m+1)} ^{m{\text{ Faktoren}}}&{\text{, wenn }}m\geq 0\\&\\\underbrace {\frac {1}{(x+1)(x+2)\ldots (x-m)}} _{|m|{\text{ Faktoren}}}&{\text{, wenn }}m<0\end{cases}}}" loading="lazy"></span></dd></dl>
<p>Dieser Ausdruck verhält sich in der Differenzenrechnung folgendermaßen:
</p>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x^{\underline {m}})=mx^{\underline {m-1}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mi>m</mi>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>m</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mrow>
<mi>m</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta (x^{\underline {m}})=mx^{\underline {m-1}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66ac262164da594d4822f456fb35c7c3811760ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.997ex; height:3.176ex;" alt="{\displaystyle \Delta (x^{\underline {m}})=mx^{\underline {m-1}}}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {}{_{a}^{b}}\sum x^{\underline {m}}\;\delta x={\begin{cases}\left[{\frac {x^{\underline {m+1}}}{m+1}}\right]_{a}^{b}&{\text{, wenn }}m\neq -1\\\left[H_{x}\right]_{a}^{b}&{\text{, wenn }}m=-1\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-OP">
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mrow class="MJX-TeXAtom-ORD">
<mpadded width="0">
<mrow class="MJX-TeXAtom-ORD">
<mphantom>
<mo>∑<!-- ∑ --></mo>
</mphantom>
</mrow>
</mpadded>
</mrow>
</mrow>
<mspace width="negativethinmathspace"></mspace>
<msubsup>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo movablelimits="false">∑<!-- ∑ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msubsup>
</mrow>
<mo><!-- --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mi>m</mi>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
<mspace width="thickmathspace"></mspace>
<mi>δ<!-- δ --></mi>
<mi>x</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<msubsup>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mrow>
<mi>m</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
<mrow>
<mi>m</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msubsup>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>, wenn </mtext>
</mrow>
<mi>m</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mrow>
<mo>[</mo>
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>]</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msubsup>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>, wenn </mtext>
</mrow>
<mi>m</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sideset {}{_{a}^{b}}\sum x^{\underline {m}}\;\delta x={\begin{cases}\left[{\frac {x^{\underline {m+1}}}{m+1}}\right]_{a}^{b}&{\text{, wenn }}m\neq -1\\\left[H_{x}\right]_{a}^{b}&{\text{, wenn }}m=-1\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5df29acd02ffcca8fd1430d5e928471c3cb29ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; margin-left: -0.387ex; width:42.828ex; height:8.843ex;" alt="{\displaystyle \sideset {}{_{a}^{b}}\sum x^{\underline {m}}\;\delta x={\begin{cases}\left[{\frac {x^{\underline {m+1}}}{m+1}}\right]_{a}^{b}&{\text{, wenn }}m\neq -1\\\left[H_{x}\right]_{a}^{b}&{\text{, wenn }}m=-1\end{cases}}}" loading="lazy"></span></li></ul>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e63458b04288bbe116a9a8037dfae0b36b2c639a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.15ex; height:2.509ex;" alt="{\displaystyle H_{n}}" loading="lazy"></span> die <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"></span>-te <a href="Harmonische_Reihe" title="Harmonische Reihe">harmonische Zahl</a> ist. Die harmonische Reihe ist somit das Gegenstück zum <a href="Logarithmus" title="Logarithmus">natürlichen Logarithmus</a>. Die Übereinstimmung geht so weit, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x\cdot H_{x}-x)=H_{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta (x\cdot H_{x}-x)=H_{x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b22c1e8f58d4a74da1d7d7395207d98acbe60669.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.23ex; height:2.843ex;" alt="{\displaystyle \Delta (x\cdot H_{x}-x)=H_{x}}" loading="lazy"></span> ebenfalls gilt.
</p><p>Fallende Fakultäten und Potenzen können stets mittels <a href="Stirling-Zahl" title="Stirling-Zahl">Stirling-Zahlen</a> erster bzw. zweiter Art ineinander umgewandelt werden:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\underline {m}}=\sum _{k}\left[{\begin{matrix}m\\k\end{matrix}}\right](-1)^{m-k}x^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mi>m</mi>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</munder>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>m</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</msup>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{\underline {m}}=\sum _{k}\left[{\begin{matrix}m\\k\end{matrix}}\right](-1)^{m-k}x^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02168d3ac8b8f164ebcff3326529243c57ae16c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.489ex; height:6.676ex;" alt="{\displaystyle x^{\underline {m}}=\sum _{k}\left[{\begin{matrix}m\\k\end{matrix}}\right](-1)^{m-k}x^{k}}" loading="lazy"></span>,</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{m}=\sum _{k}\left\{{\begin{matrix}m\\k\end{matrix}}\right\}x^{\underline {k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</munder>
<mrow>
<mo>{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>m</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>}</mo>
</mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mi>k</mi>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{m}=\sum _{k}\left\{{\begin{matrix}m\\k\end{matrix}}\right\}x^{\underline {k}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5215cf64568b28e6b81665ffc8291f2062e8fe86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.93ex; height:6.676ex;" alt="{\displaystyle x^{m}=\sum _{k}\left\{{\begin{matrix}m\\k\end{matrix}}\right\}x^{\underline {k}}}" loading="lazy"></span></dd></dl>
<p>Außerdem gilt der <a href="Binomischer_Lehrsatz" title="Binomischer Lehrsatz">binomische Lehrsatz</a> auch für fallende Fakultäten.
</p>
<div style="margin: 1em 2em 0 2em; border: 1px solid #448800; padding: 0.4em;">
<p>Beispiel zur Berechnung der Summe der ersten <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"></span> Quadratzahlen:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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}^{n}k^{2}=\sideset {}{_{0}^{n+1}}\sum x^{2}\delta x=\sideset {}{_{0}^{n+1}}\sum (x^{\underline {2}}\,+\,x^{\underline {1}})\delta x={\frac {(n+1)^{\underline {3}}}{3}}+{\frac {(n+1)^{\underline {2}}}{2}}={\frac {n(n+{\frac {1}{2}})(n+1)}{3}}}">
<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>n</mi>
</mrow>
</munderover>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-OP">
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mrow class="MJX-TeXAtom-ORD">
<mpadded width="0">
<mrow class="MJX-TeXAtom-ORD">
<mphantom>
<mo>∑<!-- ∑ --></mo>
</mphantom>
</mrow>
</mpadded>
</mrow>
</mrow>
<mspace width="negativethinmathspace"></mspace>
<msubsup>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo movablelimits="false">∑<!-- ∑ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
</mrow>
<mo><!-- --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>δ<!-- δ --></mi>
<mi>x</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-OP">
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mrow class="MJX-TeXAtom-ORD">
<mpadded width="0">
<mrow class="MJX-TeXAtom-ORD">
<mphantom>
<mo>∑<!-- ∑ --></mo>
</mphantom>
</mrow>
</mpadded>
</mrow>
</mrow>
<mspace width="negativethinmathspace"></mspace>
<msubsup>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo movablelimits="false">∑<!-- ∑ --></mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
</mrow>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mn>2</mn>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
<mspace width="thinmathspace"></mspace>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mn>1</mn>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mi>δ<!-- δ --></mi>
<mi>x</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mn>3</mn>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<munder>
<mn>2</mn>
<mo>_<!-- _ --></mo>
</munder>
</mrow>
</msup>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}k^{2}=\sideset {}{_{0}^{n+1}}\sum x^{2}\delta x=\sideset {}{_{0}^{n+1}}\sum (x^{\underline {2}}\,+\,x^{\underline {1}})\delta x={\frac {(n+1)^{\underline {3}}}{3}}+{\frac {(n+1)^{\underline {2}}}{2}}={\frac {n(n+{\frac {1}{2}})(n+1)}{3}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/732bd426f6d1d3496ce8371e6dbb9410cac156e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:87.644ex; height:7.676ex;" alt="{\displaystyle \sum _{k=0}^{n}k^{2}=\sideset {}{_{0}^{n+1}}\sum x^{2}\delta x=\sideset {}{_{0}^{n+1}}\sum (x^{\underline {2}}\,+\,x^{\underline {1}})\delta x={\frac {(n+1)^{\underline {3}}}{3}}+{\frac {(n+1)^{\underline {2}}}{2}}={\frac {n(n+{\frac {1}{2}})(n+1)}{3}}}" loading="lazy"></span>.</dd></dl>
</div>
<div class="mw-heading mw-heading3"><h3 id="Produktregel_und_partielle_Summation">Produktregel und partielle Summation</h3></div>
<p>Die <a href="Produktregel" title="Produktregel">Produktregel</a> der kontinuierlichen Differentialrechnung ist in folgender Form gültig:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (u(x)v(x))=u(x)\Delta v(x)+v(x+1)\Delta u(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta (u(x)v(x))=u(x)\Delta v(x)+v(x+1)\Delta u(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/654a2597b6b723787788ac78649e1592ccf4f07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.764ex; height:2.843ex;" alt="{\displaystyle \Delta (u(x)v(x))=u(x)\Delta v(x)+v(x+1)\Delta u(x)}" loading="lazy"></span>.</dd></dl>
<p>Diese Regel lässt sich durch Einführung eines <i>Verschiebeoperators</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">E</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be1811407dea8b43727d28dbe8da7251985b03e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \mathrm {E} }" loading="lazy"></span>, definiert als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} f(x)=f(x+1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">E</mi>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} f(x)=f(x+1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb0cdfe9d298a1f00c1fccb5675cf65260745255.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.519ex; height:2.843ex;" alt="{\displaystyle \mathrm {E} f(x)=f(x+1)}" loading="lazy"></span>, kompakter ausdrücken:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (uv)=u\Delta v+\mathrm {E} v\Delta u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>u</mi>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>v</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">E</mi>
</mrow>
<mi>v</mi>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta (uv)=u\Delta v+\mathrm {E} v\Delta u}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2b32696d9b6f3a82d404c83a6214368fb4a30a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.511ex; height:2.843ex;" alt="{\displaystyle \Delta (uv)=u\Delta v+\mathrm {E} v\Delta u}" loading="lazy"></span>.</dd></dl>
<p>Die Umstellung der Terme führt zur Formel der <i>partiellen Summation</i> ähnlich der <a href="Partielle_Integration" title="Partielle Integration">partiellen Integration</a>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 u\,\Delta v=uv-\sum \mathrm {E} v\,\Delta u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∑<!-- ∑ --></mo>
<mi>u</mi>
<mspace width="thinmathspace"></mspace>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>v</mi>
<mo>=</mo>
<mi>u</mi>
<mi>v</mi>
<mo>−<!-- − --></mo>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">E</mi>
</mrow>
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum u\,\Delta v=uv-\sum \mathrm {E} v\,\Delta u}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/519749508b621e483197d208f06303723f55821f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.024ex; height:3.843ex;" alt="{\displaystyle \sum u\,\Delta v=uv-\sum \mathrm {E} v\,\Delta u}" loading="lazy"></span>.</dd></dl>
<div style="margin: 1em 2em 0 2em; border: 1px solid #448800; padding: 0.4em;">
<p>Beispiel zur Berechnung der Summe <span class="mwe-math-element mwe-math-element-inline"><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}^{n}k2^{k}}">
<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>n</mi>
</mrow>
</munderover>
<mi>k</mi>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}k2^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc00e3eb31fb5f8d00290d90f363d8a0921bec68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:7.204ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}k2^{k}}" loading="lazy"></span>:
</p><p>Hier ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)=x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u(x)=x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f905f871c20c2b32b94c2aedbac58801c99bd693.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.897ex; height:2.843ex;" alt="{\displaystyle u(x)=x}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta v(x)=2^{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta v(x)=2^{x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/668acb99269ab1308e4fe13b53b59e8ae498ea38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.636ex; height:2.843ex;" alt="{\displaystyle \Delta v(x)=2^{x}}" loading="lazy"></span>, sodass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta u(x)=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle \Delta u(x)=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b61a7ba88c65e178c8f904ccb17df3079726167c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.665ex; height:2.843ex;" alt="{\displaystyle \Delta u(x)=1}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)=2^{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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</msup>
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<annotation encoding="application/x-tex">{\displaystyle v(x)=2^{x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cca0dfb4e3bebd5780d38453b8d9ccbe9898acee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.7ex; height:2.843ex;" alt="{\displaystyle v(x)=2^{x}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} v(x)=2^{x+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">E</mi>
</mrow>
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
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</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} v(x)=2^{x+1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92c31adff0a2355101cf61ca16fbc58f241cd488.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.383ex; height:3.176ex;" alt="{\displaystyle \mathrm {E} v(x)=2^{x+1}}" loading="lazy"></span>.
</p><p>Die Formel zur partiellen Summation ergibt:
<span class="mwe-math-element mwe-math-element-inline"><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 x2^{x}\;\delta x=x2^{x}-\sum 2^{x+1}\;\delta x=x2^{x}-2^{x+1}+C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∑<!-- ∑ --></mo>
<mi>x</mi>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msup>
<mspace width="thickmathspace"></mspace>
<mi>δ<!-- δ --></mi>
<mi>x</mi>
<mo>=</mo>
<mi>x</mi>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mo>∑<!-- ∑ --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
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</msup>
<mspace width="thickmathspace"></mspace>
<mi>δ<!-- δ --></mi>
<mi>x</mi>
<mo>=</mo>
<mi>x</mi>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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</msup>
<mo>−<!-- − --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
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</msup>
<mo>+</mo>
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum x2^{x}\;\delta x=x2^{x}-\sum 2^{x+1}\;\delta x=x2^{x}-2^{x+1}+C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96e5bc964a5fb78116778614dd9b8af67bae4188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:49.88ex; height:3.843ex;" alt="{\displaystyle \sum x2^{x}\;\delta x=x2^{x}-\sum 2^{x+1}\;\delta x=x2^{x}-2^{x+1}+C}" loading="lazy"></span>.
</p><p>Dies führt schließlich zur Lösung:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{k=0}^{n}k2^{k}&=\sideset {}{_{0}^{n+1}}\sum x2^{x}\;\delta x\\&=\left[x2^{x}-2^{x+1}\right]_{0}^{n+1}\\&=(n-1)2^{n+1}+2\end{aligned}}}">
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<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
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<mi>x</mi>
<msup>
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<mo>−<!-- − --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
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<mn>1</mn>
<mo stretchy="false">)</mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=0}^{n}k2^{k}&=\sideset {}{_{0}^{n+1}}\sum x2^{x}\;\delta x\\&=\left[x2^{x}-2^{x+1}\right]_{0}^{n+1}\\&=(n-1)2^{n+1}+2\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44f787fe7be7a3ccb774719d388f1cd5f97ee6d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:27.253ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\sum _{k=0}^{n}k2^{k}&=\sideset {}{_{0}^{n+1}}\sum x2^{x}\;\delta x\\&=\left[x2^{x}-2^{x+1}\right]_{0}^{n+1}\\&=(n-1)2^{n+1}+2\end{aligned}}}" loading="lazy"></span></dd></dl>
</div>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Koeffizienten_f%C3%BCr_Differenzenquotienten" title="Koeffizienten für Differenzenquotienten">Koeffizienten für Differenzenquotienten</a></li>
<li><a href="Differenzengleichung" title="Differenzengleichung">Differenzengleichung</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Alexander_Ossipowitsch_Gelfond" title="Alexander Ossipowitsch Gelfond">A. O. Gelfond</a>: <i>Differenzenrechnung.</i> Dt. Verlag d. Wiss., Berlin, 1958</li>
<li>Ronald Graham u. a.: <i><a href="Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics</a>.</i> Addison-Wesley, Upper Saddle River 2008, ISBN 0-201-55802-5</li>
<li><a href="Niels_Erik_N%C3%B8rlund" title="Niels Erik Nørlund">N. E. Nörlund</a>: <i>Vorlesungen über Differenzenrechnung.</i> Springer-Verlag, Berlin, 1924; Reprint Chelsea, New York, 1954</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://homepages.math.uic.edu/~kauffman/DCalc.pdf">Brian Hamrick: Discrete Calculus</a> (PDF, 70 kB)</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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