real analysis - Determine if this series $sumlimits_{n=1}^infty frac{(n!)^2}{(2n)!}$ converges or diverges, and prove your answer?

Determine if this series

$$\sum\limits_{n=1}^\infty \frac{(n!)^2}{(2n)!}$$ converges or diverges, and prove your answer?

I've been able to prove similar problems, but I'm confused now that there's a factorial involved. Can someone help me out here?

Answer

Let $a_n$ be the $n$=th term. We use the Ratio Test, and calculate $\lim_{n\to\infty} \frac{a_{n+1}}{a_n}$. We have

$$\frac{a_{n+1}}{a_n}=\frac{ \frac{((n+1)!)^2}{(2n+2)!}}{\frac{(n!)^2}{(2n)!}}=\frac{((n+1)!)^2 (2n)!}{(n!)^2(2n+2)!}$$
Now we start to simplify. Note that $\frac{(n+1)!}{n!}=n+1$ and $\frac{(2n)!}{(2n+2)!}=\frac{1}{(2n+1)(2n+2)}$, so our ratio simplifies to
$$\frac{(n+1)^2}{(2n+1)(2n+2)},$$
which further simplifies to
$$\frac{n+1}{2(2n+1)}.$$
Now find the limit as $n\to\infty$, perhaps by dividing top and bottom by $n$. The limit is $\frac{1}{4}\lt 1$, so we have convergence.