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