Originally Posted by

**arbolis** I've to show whether the following infinite series converges or not. The problem is that I tried to prove it's divergent by induction, by the root test and by trying to bound it from below. But I couldn't reach any conclusion.

Here it comes : $\displaystyle \sum_{n=1}^{+\infty} \frac{n!}{n^n}$. I'm sure it's a famous one. It's very similar to the one that defines $\displaystyle e^n$ (inversed).

I'd like you to tell me what test should I try. Then if I'm not able to get a conclusion, I'll ask for further help. Thanks in advance.