TITLE: Why does $\sum_{n=2}^\infty \frac{1}{\ln(n!)}$ diverge?
QUESTION [5 upvotes]: $$\sum_{n=2}^\infty \frac{1}{\ln(n!)}$$

I tried by comparing it to $\sum_{n=1}^\infty \frac{1}{n}$ but i seem to fail. 
I think I need to compare with series that are smaller and diverge. Help.

REPLY [16 votes]: Hint. One may observe that
$$
n! \leq n^n,\quad n\geq2,
$$ giving
$$
\ln(n!) \leq \ln(n^n)=n\ln n
$$ and, for $N\geq2$,

$$
\sum_{n=2}^N \frac{1}{\ln(n!)} \geq \sum_{n=2}^N \frac{1}{n\ln n}
$$ 

then let $N \to \infty$ to conclude.