## Calculus (3rd Edition)

The series $\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$ converges.
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.