Answer
Diverges
Work Step by Step
Given
$$\sum_{n=1}^{\infty}\left(\frac{n}{2}\right)^{n} \frac{1}{n !}$$
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}{2}\right)^{n+1} \frac{1}{(n+1) !} \left(\frac{2}{n}\right)^{n} n ! \\
&=\lim _{n \rightarrow \infty} \frac{\ln (n+1)}{\ln n} \lim _{n \rightarrow \infty} \frac{n^{3 / 2}}{(n+1)^{3 / 2}} \\
&= \frac{1}{2}\lim _{n \rightarrow \infty} \frac{(n+1)^{n}}{n^{n}}\\
&= \frac{1}{2}\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n\\
&=\frac{e}{2}>1
\end{align*}
Thus the given series diverges.
