TITLE: The series $\sum_{n=2}^{\infty }\frac{1}{\log (n!)}$
QUESTION [8 upvotes]: I'm trying to find out whether this Series
$\sum_{n=2}^{\infty } a_{n}$ converges or not when
$$a_{n}=\frac{1}{\log (n!)}$$
I tried couple of methods, among them: d'Alembert $\frac{a_{n+1}}{a_{n}}$,   Cauchy condensation test $\sum_{n=2}^{\infty } 2^{n}a_{2^n}$, and they both didn't work for me.
Edit: I can't use stirling, and integral.
Thank you

REPLY [6 votes]: You both gave me an Idea:
We know that from $n=2$
$n!< n^{n}$, so $\frac{1}{ n\log n}<\frac{1}{\log  n!}$
and now from Cauchy Condensation $\frac{1}{ n\log 2}$ is obivously  diverges and we're done.