Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{n^{4}}{n !}$$
By using the Ratio test
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{(n+1)^{4}}{(n+ 1) !} \frac{n !}{n^{4}} \\
&=\lim _{n \rightarrow \infty} \frac{(n+1)^{3}}{n^4}\\
&=0<1
\end{align*}
Thus, the given series converges.
