请输入您要查询的字词:

 

单词 PrimeHarmonicSeriesDivergesChebyshevsProof
释义

prime harmonic series diverges - Chebyshev’s proof


Theorem. p prime1p diverges.

Proof. (Chebyshev, 1880)
Consider the product

pn(1-1p)-1

Since (1-1p)-1=1+1p+1p2+, we have

pn(1-1p)-1=(1+12+122+)(1+13+132+)(1+15+152+)

So for each mn, if we expand the above product, 1m will be a term. Thus

pn(1-1p)-1x=1n1x

Taking logarithms, we have

pn-ln(1-1p)lnx=1n1x

But ln(1-u)=-u-u22-u33-, so

-ln(1-1p)=1p+12p2+13p3+1p+1p2+1p3+2p

Hence

pn2ppn(-ln(1-1p))lnx=1n1x

and thus

pn1p12lnx=1n1x

But the latter series diverges, and the result follows.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 12:16:59