Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{n!}{n^n}$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{(n+1)!}{(n+1)^{n+1}}\frac{n^n}{n!}\right|\\
&= \lim _{n \rightarrow \infty} \left(\frac{n}{n+1}\right)^n\\
&= \lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{-n}\\
&=\frac{1}{e}<1
\end{align*}
Thus the series converges.
