For which p the series converge? $$\sum_{n=0}^{\infty}\left(\frac{1}{n!}\right)^{p}$$
Please verify answer below

The series diverges for $p\le0$, since the terms do not approach $0$.  For $p>0$, the $n$-th root of the $n$-th term is
$$
(n!)^{-p/n} = e^{-p(\log n + O(1))} \sim Cn^{-p},
$$
which approaches zero.  Therefore it converges by the root test for all $p>0$.
