# 條件收斂

## 詳細定義

### 條件收斂的級數

$\sum_{n=1}^\infty a_n = C,$

$\sum_{n=1}^\infty | a_n | = \infty ,$

### 條件收斂的廣義積分

$\int_{a}^{+\infty} f(x) \mathrm{d}x = \lim_{b\to +\infty} \int_{a}^b f(x) \mathrm{d}x$

$\int_{a}^{+\infty} |f(x)| \mathrm{d}x = \lim_{b\to +\infty} \int_{a}^b |f(x)| \mathrm{d}x$

## 例子

### 無窮級數

$A_{h} = 1 - \frac12 + \frac13 - \frac14 + \cdots = \sum_{n} \frac{(-1)^{n+1}}{n}$

$\sum_{n=1}^\infty \frac{1}{n} = \infty.$

### 廣義積分

$I = \int_{1}^{+\infty} \frac{\sin x}{x} \mathrm{d}x$

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

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

$\int_{1}^{+\infty} \frac{\sin x}{x} \mathrm{d}x = \lim_{a\to +\infty}\int_{1}^{a} \frac{\sin x}{x} \mathrm{d}x =\cos 1 - \lim_{a\to +\infty} \frac{\cos a}{a} - \lim_{a\to +\infty}\int_{1}^{a} \frac{\cos x}{x^2} \mathrm{d}x = \cos 1 - \int_{1}^{+\infty} \frac{\cos x}{x^2} \mathrm{d}x$

$I_k = \int_{k\pi}^{(k+1)\pi} \bigg|\frac{\sin x}{x} \bigg| \mathrm{d}x \geqslant \int_{k\pi}^{(k+1)\pi} \frac{|\sin x|}{(k+1)\pi} \mathrm{d}x = \frac{2}{(k+1)\pi} = \frac{2}{\pi}\cdot \frac{1}{k+1}$

$I_s = \int_{1}^{+\infty} \bigg|\frac{\sin x}{x} \bigg| \mathrm{d}x \geqslant \sum_{k=1}^{+\infty} I_k \geqslant \frac{2}{\pi}\cdot \sum_{k=1}^{+\infty} \frac{1}{k+1} = +\infty$

## 相關定理

• 黎曼級數定理：假設$\sum_{n=1}^\infty a_n$是一個條件收斂的無窮級數。對任意的一個實數$C$，都存在一種從自然數集合到自然數集合的排列$\sigma : \, \, n \mapsto \sigma (n)$，使得
$\sum_{n=1}^\infty a_{\sigma (n)} = C.$

$\sum_{n=1}^\infty a_{\sigma' (n)} = \infty.$

## 參考來源

1. ^ J. A. Fridy. Introductory analysis: the theory of calculus. Gulf Professional Publishing. 2000. ISBN 9780122676550.
2. ^ 2.0 2.1 清華大學數學科學系. 《微積分》. 北京: 清華大學出版社有限公司. 2003. ISBN 9787302069171.
3. ^ 3.0 3.1 S. Ponnusamy. Foundations of mathematical analysis. Springer. 2012. ISBN 9780817682927.