Why does $\sum_{n=2}^\infty \frac{1}{\ln(n!)}$ diverge?

Question

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

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.

Answer 1

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.