> 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/zh-cn/shu-xue/09-integrals-sums-and-limits.md).
# 积分、求和与极限
## 积分
可以使用以下方式添加积分表达式： `\int_{lower}^{upper}` 命令。
注意，积分表达式在行内数学模式和显示数学模式下看起来可能会有些不同。
| LaTeX 代码 | 输出 |
| ---------------------------------- | -------------------------------------------------------------------------------------- |
| `文本中的积分 \(\int_{a}^{b} x^2 \,dx\)` |  |
| `\[ \int_{a}^{b} x^2 \,dx \]` |  |
## 多重积分
要获得双重/三重/多重积分和闭合积分，你必须使用 `amsmath` 和 `esint` （用于闭合积分）宏包。
| LaTeX 代码 | 输出 |
| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- |
| `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*}
` |  |
| `latex
\[
\oint_V f(s) \,ds
\]
` |  |
## 求和与乘积
与积分类似，可以使用以下方式添加求和表达式： `\sum_{lower}^{upper}` 命令。
| LaTeX 代码 | 输出 |
| ----------------------------------------- | ----------------------------------------------------------------------------------------- |
| `文本中的求和 $\sum_{n=1}^{\infty} 2^{-n} = 1$` |  |
| `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` |  |
类似地，你可以使用以下方式获得由一系列因子组成的乘积表达式： `\prod_{lower}^{upper}` 命令。
| LaTeX 代码 | 输出 |
| ------------------------------- | ------------------------------------------------------------------------------------------- |
| `文本中的乘积 $\prod_{i=a}^{b} f(i)$` |  |
| `\[ \prod_{i=a}^{b} f(i) \]` |  |
## 极限
可以使用以下方式添加极限表达式： `\lim_{lower}` 命令。
| LaTeX 代码 | 输出 |
| --------------------------------- | --------------------------------------------------------------------------------- |
| `文本中的极限 $\lim_{x\to\infty} f(x)$` |  |
| `\[ \lim_{x\to\infty} f(x) \]` |  |
## 示例
下面是一个 LaTeX 示例文档：
```latex
\documentclass{article}
\title{积分、求和与极限}
\author{Overleaf}
\date{}
\usepackage{amsmath}
\begin{document}
\maketitle
\section{积分}
文本中的积分 \(\int_{a}^{b} x^2 dx\)。
\\medskip
显示形式下的同一积分：
\[
\int_{a}^{b} x^2 \,dx
\]
以及多重积分：
\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{求和与乘积}
文本中的求和 \(\sum_{n=1}^{\infty} 2^{-n} = 1\)。
显示形式下的同一求和：
\[
\sum_{n=1}^{\infty} 2^{-n} = 1
\]
文本中的乘积 \(\prod_{i=a}^{b} f(i)\)。
显示形式下的同一乘积：
\[
\prod_{i=a}^{b} f(i)
\]
\section{极限}
文本中的极限 \(\lim_{x\to\infty} f(x)\)。
显示形式下的同一极限：
\[
\lim_{x\to\infty} f(x)
\]
\end{document}
```
[在 Overleaf 中打开此示例](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)
## 进一步阅读
更多信息请参见
* [数学表达式](/latex/zh-cn/shu-xue/01-mathematical-expressions.md)
* [下标和上标](/latex/zh-cn/shu-xue/02-subscripts-and-superscripts.md)
* [分数和二项式](/latex/zh-cn/shu-xue/05-fractions-and-binomials.md)
* [数学模式中的显示样式](/latex/zh-cn/shu-xue/10-display-style-in-math-mode.md)
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