Convergence of Series Involving Logarithms and Reciprocals

[Mr Davis 97]

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...

 

[fresh_42]

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.