Answer
The series $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$ converges.
Work Step by Step
We apply the ratio test as follows, $a_n=\frac{n!}{(2n)!}$
$$
\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty}\frac{(n+1)!/((2(n+1))!)}{n!/(2n)!}\\
=\lim _{n \rightarrow \infty}\frac{n+1}{(2(n+)(2n+1)}=0\lt 1
$$
Hence the series $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$ converges.
