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Does the following series converge or diverge?

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

Hint: One way to decide this is to use Stirling's approximation for $n!$, which says that:

$$\lim_{n\to\infty}\frac{n!}{n^n e^{-n}\sqrt{2\pi n}}=1$$

It converges.

It diverges.