Convergency of series The task is to prove, that series: $$\sum_{n=2}^\infty \frac{1}{log^2(n!)}$$ converges. Unfortunately I only managed to deal with showing that Limit of the summand is equal to 0, what is pretty obvious, but still, I am truly clueless, how to show that it converges. I am not asking for solution, but only clues/hints and other stuff like that. I would be truly grateful for any help.

**Hint:** you can easily show that $\ln(n!) \geq c n\ln n$ for some absolute constant $c >0$. From there, $\frac{1}{\ln^2 n!} \leq \frac{1}{c^2 n^2\ln^2 n}$, and conclude by theorems of comparison.