Divergent infinite series n!e^n/n^n - simpler prove of divergence?

by Skyfall   Last Updated January 11, 2019 11:20 AM


Sum from n=1 to infinity of n!*e^n/n^n

Where e is Euler's number.

Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus teacher told us that it diverges but he doesn't see any elementary way or simple test to prove it. He told us something about Euler's Gamma function and said that he knows it diverges because of the Stirling's formula but this is not quite our level yet.

I tried to think about it myself but I established only that this is some kind of an edge case. We could generalize the problem to series of n!*a^n/n^n. Then, from the ratio test we know that it converges for a e. For a=e the test is inconclusive. I've also tried to compare it with various divergent series, but nothing worked...

Does anybody have any idea for a simple solution?