# Series involving log and 1/n

## Homework Statement

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

## 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
Mentor
2021 Award
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.