> For the complete documentation index, see [llms.txt](https://overleaf-pro.ayaka.space/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://overleaf-pro.ayaka.space/latex/de/mathematik/09-integrals-sums-and-limits.md). # Integrale, Summen und Grenzwerte ## Integrale Ein Integral-Ausdruck kann mit dem hinzugefügt werden `\int_{lower}^{upper}` Befehl zu interpretieren ist. Beachten Sie, dass der Integral-Ausdruck im Inline- und im Display-Mathemodus etwas unterschiedlich aussehen kann. | LaTeX-Code | Ausgabe | | -------------------------------------------- | -------------------------------------------------------------------------------------- | | `Integral \(\int_{a}^{b} x^2 \,dx\) im Text` | ![Example of an integral inside text](/files/f0fe9408d762b900d8a01d4c76f1574f2e44ceee) | | `\[ \int_{a}^{b} x^2 \,dx \]` | ![Example of an integral](/files/8f5994fad5b2d02e4da9f7b371fafeb8414e9c32) | ## Mehrfachintegrale Um Doppel-/Dreifach-/Mehrfachintegrale und Umlaufintegrale zu erhalten, müssen Sie `amsmath` und `esint` (für Umlaufintegrale) Pakete. | LaTeX-Code | Ausgabe | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- | | `latex
\begin{gather*}
\iint_V \mu(u,v) \,du\,dv
\\
\iiint_V \mu(u,v,w) \,du\,dv\,dw
\\
\iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw
\\
\idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k
\end{gather*}
` | ![Example of multiple integrals](/files/61958fc7a9505f539ed179ba8706a7c3999b9caa) | | `latex
\[
\oint_V f(s) \,ds
\]
` | ![Example of integrals](/files/552d0a397eec8a89f39890f2c0660f5322a57470) | ## Summen und Produkte Wie beim Integral kann ein Summenausdruck mit dem hinzugefügt werden `\sum_{lower}^{upper}` Befehl zu interpretieren ist. | LaTeX-Code | Ausgabe | | ------------------------------------------------ | ----------------------------------------------------------------------------------------- | | `Summe $\sum_{n=1}^{\infty} 2^{-n} = 1$ im Text` | ![Example of sum expression inside text](/files/d006c96b13cbe9199602e4f6f7bc964d2eb0cba4) | | `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` | ![Example of sum expression](/files/d6728ba14b34cac97ef5a30ed4a0fca0a92a61e1) | Auf ähnliche Weise können Sie einen Ausdruck mit dem Produkt einer Folge von Faktoren mithilfe des `\prod_{lower}^{upper}` Befehl zu interpretieren ist. | LaTeX-Code | Ausgabe | | ---------------------------------------- | ------------------------------------------------------------------------------------------- | | `Produkt $\prod_{i=a}^{b} f(i)$ im Text` | ![Example of limit expression inside text](/files/c142ce565c23bcb3b3689ee71793245587a1ad75) | | `\[ \prod_{i=a}^{b} f(i) \]` | ![Example of limit expression](/files/29c4013ad7e14c514a6b02ab6869ca7dc482223d) | ## Grenzwerte Ein Grenzwert-Ausdruck kann mit dem hinzugefügt werden `\lim_{lower}` Befehl zu interpretieren ist. | LaTeX-Code | Ausgabe | | -------------------------------------------- | --------------------------------------------------------------------------------- | | `Grenzwert $\lim_{x\to\infty} f(x)$ im Text` | ![Example of limits inside text](/files/314a06c07d37bb29f4a9e3d4173a7734e1cc598d) | | `\[ \lim_{x\to\infty} f(x) \]` | ![Example of limits](/files/a6dce6d28b9df2918422a073980654caf57745e7) | ## Beispiele Hier ist ein Beispieldokument in LaTeX: ```latex \documentclass{article} \title{Integrale, Summen und Grenzwerte} \author{Overleaf} \date{} \usepackage{amsmath} \begin{document} \maketitle \section{Integrale} Integral \(\int_{a}^{b} x^2 dx\) im Text. \medskip Dasselbe Integral in der Anzeige: \[ \int_{a}^{b} x^2 \,dx \] und Mehrfachintegrale: \begin{gather*} \iint_V \mu(u,v) \,du\,dv \\ \iiint_V \mu(u,v,w) \,du\,dv\,dw \\ \iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw \\ \idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k \\ \oint_V f(s) \,ds \end{gather*} \section{Summen und Produkte} Summe \(\sum_{n=1}^{\infty} 2^{-n} = 1\) im Text. Dieselbe Summe in der Anzeige: \[ \sum_{n=1}^{\infty} 2^{-n} = 1 \] Produkt \(\prod_{i=a}^{b} f(i)\) im Text. Dasselbe Produkt in der Anzeige: \[ \prod_{i=a}^{b} f(i) \] \section{Grenzwerte} Grenzwert \(\lim_{x\to\infty} f(x)\) im Text. Derselbe Grenzwert in der Anzeige: \[ \lim_{x\to\infty} f(x) \] \end{document} ``` [Öffnen Sie dieses Beispiel in Overleaf](https://www.overleaf.com/docs?engine=lualatex\&snip_name=Integrals%2C+Sums+and+Limits\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Ctitle%7BIntegrals%2C+Sums+and+Limits%7D%0A%5Cauthor%7BOverleaf%7D%0A%5Cdate%7B%7D%0A%5Cusepackage%7Bamsmath%7D%0A%0A%5Cbegin%7Bdocument%7D%0A%0A%5Cmaketitle%0A%0A%5Csection%7BIntegrals%7D%0A%0AIntegral+%5C%28%5Cint_%7Ba%7D%5E%7Bb%7D+x%5E2+dx%5C%29+inside+text.%0A%0A%5Cmedskip%0A%0AThe+same+integral+on+display%3A%0A%5C%5B%0A++++%5Cint_%7Ba%7D%5E%7Bb%7D+x%5E2+%5C%2Cdx%0A%5C%5D%0Aand+multiple+integrals%3A%0A%5Cbegin%7Bgather%2A%7D%0A++++%5Ciint_V+%5Cmu%28u%2Cv%29+%5C%2Cdu%5C%2Cdv%0A%5C%5C%0A++++%5Ciiint_V+%5Cmu%28u%2Cv%2Cw%29+%5C%2Cdu%5C%2Cdv%5C%2Cdw%0A%5C%5C%0A++++%5Ciiiint_V+%5Cmu%28t%2Cu%2Cv%2Cw%29+%5C%2Cdt%5C%2Cdu%5C%2Cdv%5C%2Cdw%0A%5C%5C%0A++++%5Cidotsint_V+%5Cmu%28u_1%2C%5Cdots%2Cu_k%29+%5C%2Cdu_1+%5Cdots+du_k%0A%5C%5C%0A++++%5Coint_V+f%28s%29+%5C%2Cds%0A%5Cend%7Bgather%2A%7D%0A%0A%5Csection%7BSums+and+products%7D%0A%0ASum+%5C%28%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+2%5E%7B-n%7D+%3D+1%5C%29+inside+text.%0A%0AThe+same+sum+on+display%3A%0A%5C%5B%0A++++%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+2%5E%7B-n%7D+%3D+1%0A%5C%5D%0A%0AProduct+%5C%28%5Cprod_%7Bi%3Da%7D%5E%7Bb%7D+f%28i%29%5C%29+inside+text.%0A%0AThe+same+product+on+display%3A%0A%5C%5B%0A++++%5Cprod_%7Bi%3Da%7D%5E%7Bb%7D+f%28i%29%0A%5C%5D%0A%0A%5Csection%7BLimits%7D%0A%0ALimit+%5C%28%5Clim_%7Bx%5Cto%5Cinfty%7D+f%28x%29%5C%29+inside+text.%0A%0AThe+same+limit+on+display%3A%0A%5C%5B%0A++++%5Clim_%7Bx%5Cto%5Cinfty%7D+f%28x%29%0A%5C%5D%0A%0A%5Cend%7Bdocument%7D) ## Weiterführende Lektüre Weitere Informationen finden Sie unter * [Mathematische Ausdrücke](/latex/de/mathematik/01-mathematical-expressions.md) * [Tief- und Hochstellungen](/latex/de/mathematik/02-subscripts-and-superscripts.md) * [Brüche und Binome](/latex/de/mathematik/05-fractions-and-binomials.md) * [Darstellungsstil im Mathematikmodus](/latex/de/mathematik/10-display-style-in-math-mode.md) --- # Agent Instructions This documentation is published with GitBook. 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