# 调和级数

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{k^s}$

$\sum_{k=1}^\infty \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\,\!$

## 佯谬

$\frac{1}{100}\sum_{k=1}^n\frac{1}{k}.$

$d_{n+1}\,=\,\frac{(d_n+l_n)n+\frac{l_0}{2}}{n+1}\,=\,\frac{l_0\cdot n+\frac{l_0}{2}}{n+1}\,=\,\frac{l_0\cdot (n+1)-\frac{l_0}{2}}{n+1}\,=\,l_0- \frac{\frac{l_0}{2}}{n+1}$

$l_{n+1} = l_0 - d_{n+1} = \frac{\frac{l_0}{2}}{n+1}$，即$l_n = \frac{l_0}{2} \cdot \frac{1}{n}$

$l_{\mathrm{total}} = \frac{l_0}{2} \cdot \sum_{k=1}^n\frac{1}{k}$

## 发散性

### 比较审敛法

$\sum_{k=1}^\infty \frac{1}{k} = 1 + \left[\frac{1}{2}\right] + \left[\frac{1}{3} + \frac{1}{4}\right] + \left[\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right] + \left[\frac{1}{9}+\cdots\right.$
$\quad\ \ge \sum_{k=1}^\infty 2^{-\lceil \log_2 k \rceil}\,\!$
$= 1 + \left[\frac{1}{2}\right] + \left[\frac{1}{4} + \frac{1}{4}\right] + \left[\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right] + \left[\frac{1}{16}+\cdots\right.\,\!$
$= 1 +\ \frac{1}{2}\ + \qquad\frac{1}{2} \ \quad+ \ \qquad\quad\frac{1}{2}\qquad\ \quad \ + \ \quad\ \cdots \,\!\;=\;\; \infty.$

### 积分判别法

$\sum_{n=1}^k \, \frac{1}{n} \;>\; \int_1^{k+1} \frac{1}{x}\,dx \;=\; \ln(k+1).$

### 反证法

$\lim _{n \to \infty} S_{2n}-S_n=0$

## 发散率

$\sum_{n=1}^k\,\frac{1}{n} \;=\; \ln k + \gamma + \varepsilon_k$

## 部分和

$H_n = \sum_{k = 1}^n \frac{1}{k},\!$

## 相关级数

### 交错调和级数

$\sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} \;=\; 1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots$

$1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots \;=\; \ln 2.$

$\sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} \;\;=\;\; 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \cdots \;\;=\;\; \frac{\pi}{4}.$

### 广义调和级数

$\sum_{n=0}^{\infty}\frac{1}{an+b}.\!$

### P-级数

$\sum_{n=1}^{\infty}\frac{1}{n^p},\!$

### φ-级数

$\limsup_{u\to 0^{+}}\frac{\varphi(\frac{u}{2})}{\varphi(u)}< \frac{1}{2}$

### 随机调和级数

$\sum_{n=1}^{\infty}\frac{s_{n}}{n},\!$

## 参考

1. ^ George L. Hersey, Architecture and Geometry in the Age of the Baroque, p 11-12 and p37-51.
2. ^ 2.0 2.1 Graham, Ronald; Knuth, Donald E.; Patashnik, Oren, Concrete Mathematics 2nd, Addison-Wesley, 258–264, 1989, ISBN 978-0-201-55802-9
3. ^ Sharp, R.T., Problem 52: Overhanging dominoes, Pi Mu Epsilon Journal, 1954: 411–412
4. ^ Sequence A082912 in the On-Line Encyclopedia of Integer Sequences
5. ^ http://mathworld.wolfram.com/HarmonicNumber.html
6. ^ Art of Problem Solving: "General Harmonic Series"
7. ^ "Random Harmonic Series", American Mathematical Monthly 110, 407-416, May 2003
8. ^ Schmuland's preprint of Random Harmonic Series
9. ^ Weisstein, Eric W. “Infinite Cosine Product Integral.” From MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/InfiniteCosineProductIntegral.html accessed 11/14/2010
10. ^ Nick's Mathematical Puzzles: Solution 72