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调和级数

维基百科，自由的百科全书

${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k))=1+{\frac {1}{2))+{\frac {1}{3))+{\frac {1}{4))+\cdots \,\!}$

佯谬

${\displaystyle {\frac {1}{100))\sum _{k=1}^{n}{\frac {1}{k)).}$

${\displaystyle d_{n+1}\,=\,{\frac {(d_{n}+l_{n})n+{\frac {l_{0)){2))}{n+1))\,=\,{\frac {l_{0}\cdot n+{\frac {l_{0)){2))}{n+1))\,=\,{\frac {l_{0}\cdot (n+1)-{\frac {l_{0)){2))}{n+1))\,=\,l_{0}-{\frac {\frac {l_{0)){2)){n+1))}$

${\displaystyle l_{n+1}=l_{0}-d_{n+1}={\frac {\frac {l_{0)){2)){n+1))}$，即${\displaystyle l_{n}={\frac {l_{0)){2))\cdot {\frac {1}{n))}$

${\displaystyle l_{\mathrm {total} }={\frac {l_{0)){2))\cdot \sum _{k=1}^{n}{\frac {1}{k))}$

发散性

比较审敛法

${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k))=1+\left[{\frac {1}{2))\right]+\left[{\frac {1}{3))+{\frac {1}{4))\right]+\left[{\frac {1}{5))+{\frac {1}{6))+{\frac {1}{7))+{\frac {1}{8))\right]+\left[{\frac {1}{9))+\cdots \right.}$
${\displaystyle \quad \ \geq \sum _{k=1}^{\infty }2^{-\lceil \log _{2}k\rceil }\,\!}$
${\displaystyle =1+\left[{\frac {1}{2))\right]+\left[{\frac {1}{4))+{\frac {1}{4))\right]+\left[{\frac {1}{8))+{\frac {1}{8))+{\frac {1}{8))+{\frac {1}{8))\right]+\left[{\frac {1}{16))+\cdots \right.\,\!}$
${\displaystyle =1+\ {\frac {1}{2))\ +\qquad {\frac {1}{2))\ \quad +\ \qquad \quad {\frac {1}{2))\qquad \ \quad \ +\ \quad \ \cdots \,\!\;=\;\;\infty .}$

积分判别法 (The integral test)

${\displaystyle \sum _{n=1}^{k}\,{\frac {1}{n))\;>\;\int _{1}^{k+1}{\frac {1}{x))\,dx\;=\;\ln(k+1).}$

反证法

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

发散率

${\displaystyle \sum _{n=1}^{k}\,{\frac {1}{n))\;=\;\ln k+\gamma +\varepsilon _{k))$

部分和

${\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k)),\!}$

相关级数

交错调和级数

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

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

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

广义调和级数

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{an+b)).\!}$

${\displaystyle p}$-级数

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{p))},\!}$

${\displaystyle \varphi }$-级数

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

随机调和级数

${\displaystyle \sum _{n=1}^{\infty }{\frac {s_{n)){n)),\!}$

参考

1. ^ George L. Hersey, Architecture and Geometry in the Age of the Baroque, p 11-12 and p37-51.
2. 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. ^ 存档副本. [2011-01-16]. （原始内容存档于2013-05-16）.
6. ^ Art of Problem Solving: "General Harmonic Series"页面存档备份，存于互联网档案馆
7. ^ "Random Harmonic Series", American Mathematical Monthly 110, 407-416, May 2003
8. ^ Schmuland's preprint of Random Harmonic Series (PDF). [2011-01-16]. （原始内容 (PDF)存档于2011-06-08）.
9. ^ Weisstein, Eric W. “Infinite Cosine Product Integral.” From MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/InfiniteCosineProductIntegral.html页面存档备份，存于互联网档案馆） accessed 11/14/2010
10. ^ Nick's Mathematical Puzzles: Solution 72. [2011-01-16]. （原始内容存档于2010-09-28）.

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