calculus - The series $sum_{n=2}^{infty }frac{1}{log (n!)}$

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

Answer

Hint: You can use

$$a_n = \frac{1}{\log n!} = \frac{1}{\sum_{k=1}^n \log k} \geq \frac{1}{n \log n}.$$
Then use the Cauchy condensation test...