Series involving log and 1/n

  • #1
1,462
44

Homework Statement


Show that ##\sum_{n=1}^{\infty}\frac{\log (1+1/n)}{n}## converges.

Homework Equations




The Attempt at a Solution


If I take for granted the inequality ##\log (1+1/n) < 1/n##, I can easily show that this converges. My problem is is that I am not seeing how to prove convergence another way...
 

Answers and Replies

  • #2
14,399
11,712
Not sure why you want to do it another way. But in any case, the first question is: What is ##\log(.)##? Is it a limit, a series, the solution of a functional equation, an isomorphism, an integral, the solution of a differential equation, or just the solution to ##e^x=c\,?## So any approach depends on what you have. Which is it? And it might happen, that the first step will be to deduce some appropriate boundary and you'll end up at what you wanted to avoid.
 

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