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 August 18th, 2010, 09:05 PM #1 Senior Member   Joined: Dec 2006 Posts: 167 Thanks: 3 The limit of a series of functions Find $\lim_{x \to \infty} \sum_{n= 0}^{\infty} \frac {x a^n} {(x+a^n)^2}$
 August 18th, 2010, 09:56 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: The limit of a series of functions I'm sure the right answer is 1/log(a), but I can't prove it.
 August 18th, 2010, 10:10 PM #3 Senior Member   Joined: Dec 2006 Posts: 167 Thanks: 3 Re: The limit of a series of functions I can prove that $\lim_{x \to \infty} \frac {x}{\ln x} \sum_{n=0}^{\infty} \frac{1}{x+a^n} = \frac{1}{\ln a}$ And $\; \frac { \left(\sum_{n=0}^{\infty} \frac{x}{x+a^n}\right)'}{( \ln x)#39;} = \sum_{n=0}^{\infty} \frac{x a^n}{(x+a^n)^2}$ Also I know that $\overline{\lim}_{x \to \infty}\sum_{n=0}^{\infty} \frac{x a^n}{(x+a^n)^2}$ is finite

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