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 January 30th, 2013, 12:43 PM #1 Newbie   Joined: Jan 2013 Posts: 13 Thanks: 0 how can solve this series? Hi all, I have to study the convergence of this series: $\sum \limits_{k=2}^\infty $$\frac {1}{ln(k)}- \frac {1}{ln(k+1)}$$$ I know (from wolfram) that for the comparison test the series converges (in particularly to 1,4427) I know that if I find a new series, b(k), such that for all k b(k)>a(k) then if b(k) converges also a(k) converges.. but what is b(k)? and what is the right method to find it? thanks to all, and I'm sorry for my bad english
 January 30th, 2013, 03:35 PM #2 Member     Joined: Mar 2011 Posts: 49 Thanks: 5 Re: how can solve this series? Hello! The partial sum for this serie is $S_n=\sum_{k=2}^n $$\frac{1}{\ln k} - \frac{1}{\ln(k+1)}$$$ by telscoping : $S_n=\frac{1}{\ln 2} - \frac{1}{\ln(n+1)} \)$ The sum of the serie is: $\lim_{n \to +\infty} S_n=\frac{1}{\ln 2}$

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