Problem Two
Following the steps below, show that the following series is convergent.
$\sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!}$
\begin{itemize}
\item First, show (using $(m+1)! = (m+1)m!$ for any $m$) that
$(2(n+1))! = (2n+2)(2n+1)(2n)!$
\item Thus, show that
$\lim_{n \to \infty} \frac{((n+1)!)^2 (2n)!}{(2(n+1))! (n!)^2} = \lim_{n \to \infty} \frac{(n+1)^2}{(2n+2)(2n+1)} = \frac{1}{4}$
\item What does the Ratio Test tell us about convergence or divergence of the series?
\end{itemize}
