I was wondering if there is a closed-form expression for
$$\sum_{n=0}^{\infty} \frac{x^n}{e^{-n}n^n},$$
although I expect there is none because Mathematica cannot compute it. However, from Stirling's approximation
$$n! \approx e^{-n} n^n$$
I would expect this sum to be $\approx e^x$ and, in particular, convergent everywhere.