real analysis - The convergence of $sum_{n=1}^infty (-1)^nleft(frac{n}{e}right)^nfrac{1}{n!}$

I'm trying to figure out if the $\sum_{n=1}^\infty (-1)^n\left(\frac{n}{e}\right)^n\frac{1}{n!}$ converges or not. I've tried the Leibnitz test for alternating series, but it leads to Stirling's formula and I was wondering if there's any other way so I could avoid using it. I'll be grateful for any idea.