Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty}\frac{n^{3}}{n !}$$
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)^{3}}{(n+1) !}\frac{n !}{n^{3}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{(n+1)^{3}}{(n+1)n !}\frac{n !}{n^{3}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{(n+1)^{2}}{n^3} \right|\\
&=0<1
\end{align*}
Thus the series converges.
