Consider the series $\sum n!\exp(-n^2)$.
By the ratio test, I've proven that it is convergent (unless I made a horrible mistake).
However, I'm trying to prove this using Stirling's Formula for $n!$.
I can't go any further than:
$ n!\exp(-n^2) \sim \sqrt{2\pi} \exp((n + \frac{1}{2})\ln(n) - n - n^2) $
How can I proceed? Or is there any other method without using the ratio test or Stirling's approximation? Or maybe I'm wrong and it's divergent?
Thank you for your answers!
Your expression from Stirling is good : $$n! \exp(-n^2) \sim \sqrt{2\pi} \exp\left(\left(n+ \frac{1}{2}\right)\ln(n) - n^2\right)=o\left(\exp\left(- \frac{n^2}{2} \right) \right)$$
and $\exp\left(- \frac{n^2}{2} \right) $ is the general term of a convergent series. Therefore, your series converges.