Im trying to find the radius of convergence for $\sum_{n=0}^\infty \frac{n!}{n^n}z^n.$
Applying the ratio test $\frac{C_{n+1}}{C_n}$, I simplified $\frac{n!}{n^n}$ to $\frac{n^n}{(n+1)^n}$.
On Wolfram, it says the limit of $\frac{n^n}{(n+1)^n}$ as $n$ tends to infinity is $\frac{1}{e}$.
Im trying to find $ \rho = \frac{1}{\lim \limits_{n \to \infty} \frac{C_{n+1}}{C_n}}, $
so $\rho = e$?
If not, how would I go about finding the radius of convergence? Do I need to use the squeeze theorem maybe? Thanks.