Does $\sum \frac{n!e^n}{n^n}$ converge or diverge

Question

I have tried root and ratio test but they were inconclusive. thank you very much

Answer 1

Using Stirling's formula, you should find that $$\frac{n!\,\mathrm e^n}{n^n}\sim_\infty\sqrt{2\pi n},$$ so it diverges trivially.