TITLE: Why does the series $\sum_{n=1}^∞ \ln ({n \over n+1})$ diverges? And general tips about series and the logarithm
QUESTION [5 upvotes]: Why does the series $\sum_{n=1}^∞ \ln ({n \over n+1})$ diverges? I'm looking for an answer using the comparison test, I'm just not sure what I can compare it to. 
And can I have some tips on what to look at when handling with series that have logarithms in the expression? 
Thanks in advance!

REPLY [2 votes]: $\sum_{n=1}^m \ln ({n \over n+1})
=\ln \prod_{n=1}^m ({n \over n+1})
= \ln \frac{1}{m+1}
$
and since
$\frac{1}{m+1} \to 0$
as $m \to =\infty$,
$\ln \frac{1}{m+1} \to -\infty$
as $m \to \infty$.