Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{n^{40}}{n! }$$
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} \frac{(n+1)^{40}}{(n+1)! } \frac{n! }{n^{40}}\\
&= \lim _{n \rightarrow \infty} \frac{1}{n+1}\left(1+\frac{1}{n}\right)^{40}\\
&=\lim _{n \rightarrow \infty} \frac{1}{n+1}\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{40}\\
&= 0<1
\end{align*}
Thus the series converges.
