Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{n^2 }{(2n+1)!}$$
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{(n+1)^2 }{(2n+3)!}\frac{(2n+1)!}{n^2 } \\
&=\lim _{n \rightarrow \infty} \frac{(n+1)^2 }{n^2}\lim _{n \rightarrow \infty} \frac{(2n+1)! }{(2n+3)(2n+2)(2n+1)!} \\
&=\lim _{n \rightarrow \infty} \frac{(n+1)^2 }{n^2}\lim _{n \rightarrow \infty} \frac{1 }{(2n+3)(2n+2) } \\
&=0<1
\end{align*}
Thus the series converges.
