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