# 条件收敛

## 详细定义

### 条件收敛的级数

${\displaystyle \sum _{n=1}^{\infty }a_{n}=C,}$

${\displaystyle \sum _{n=1}^{\infty }|a_{n}|=\infty ,}$

### 条件收敛的广义积分

${\displaystyle \int _{a}^{+\infty }f(x)\mathrm {d} x=\lim _{b\to +\infty }\int _{a}^{b}f(x)\mathrm {d} x}$

${\displaystyle \int _{a}^{+\infty }|f(x)|\mathrm {d} x=\lim _{b\to +\infty }\int _{a}^{b}|f(x)|\mathrm {d} x}$

## 例子

### 无穷级数

${\displaystyle A_{h}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots =\sum _{n}{\frac {(-1)^{n+1}}{n}}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=\infty .}$

### 广义积分

${\displaystyle I=\int _{1}^{+\infty }{\frac {\sin x}{x}}\mathrm {d} x}$

${\displaystyle \int _{1}^{a}{\frac {\sin x}{x}}\mathrm {d} x=\cos 1-{\frac {\cos a}{a}}-\int _{1}^{a}{\frac {\cos x}{x^{2}}}\mathrm {d} x}$

${\displaystyle {\Bigg |}\int _{A}^{B}{\frac {\cos x}{x^{2}}}\mathrm {d} x{\Bigg |}\leqslant \int _{A}^{B}{\frac {|\cos x|}{x^{2}}}\mathrm {d} x\leqslant \int _{A}^{B}{\frac {1}{x^{2}}}\mathrm {d} x\leqslant {\frac {1}{A}}}$

${\displaystyle \int _{1}^{+\infty }{\frac {\sin x}{x}}\mathrm {d} x=\lim _{a\to +\infty }\int _{1}^{a}{\frac {\sin x}{x}}\mathrm {d} x=\cos 1-\lim _{a\to +\infty }{\frac {\cos a}{a}}-\lim _{a\to +\infty }\int _{1}^{a}{\frac {\cos x}{x^{2}}}\mathrm {d} x=\cos 1-\int _{1}^{+\infty }{\frac {\cos x}{x^{2}}}\mathrm {d} x}$

${\displaystyle I_{k}=\int _{k\pi }^{(k+1)\pi }{\bigg |}{\frac {\sin x}{x}}{\bigg |}\mathrm {d} x\geqslant \int _{k\pi }^{(k+1)\pi }{\frac {|\sin x|}{(k+1)\pi }}\mathrm {d} x={\frac {2}{(k+1)\pi }}={\frac {2}{\pi }}\cdot {\frac {1}{k+1}}}$

${\displaystyle I_{s}=\int _{1}^{+\infty }{\bigg |}{\frac {\sin x}{x}}{\bigg |}\mathrm {d} x\geqslant \sum _{k=1}^{+\infty }I_{k}\geqslant {\frac {2}{\pi }}\cdot \sum _{k=1}^{+\infty }{\frac {1}{k+1}}=+\infty }$

## 相关定理

• 黎曼级数定理：假设${\displaystyle \sum _{n=1}^{\infty }a_{n}}$是一个条件收敛的无穷级数。对任意的一个实数${\displaystyle C}$，都存在一种从自然数集合到自然数集合的排列${\displaystyle \sigma :\,\,n\mapsto \sigma (n)}$，使得
${\displaystyle \sum _{n=1}^{\infty }a_{\sigma (n)}=C.}$

${\displaystyle \sum _{n=1}^{\infty }a_{\sigma '(n)}=\infty .}$

## 参考来源

1. ^ J. A. Fridy. Introductory analysis: the theory of calculus. Gulf Professional Publishing. 2000. ISBN 9780122676550.
2. 清华大学数学科学系. 《微积分》. 北京: 清华大学出版社有限公司. 2003. ISBN 9787302069171.
3. S. Ponnusamy. Foundations of mathematical analysis. Springer. 2012. ISBN 9780817682927.