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