convergence and divergence of series with a summand Does $\sum_{n \in \mathbb N}\frac{n!}{(2n)!}$ converge?

I tried the ratio test however confused especially with the summand before it
Let $a_k=\frac{k!}{(2k)!}$. For the Ratio Test, we examine 
$$\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|.\tag{1}$$
In our case, we have
$$\left|\frac{a_{n+1}}{a_n}\right|=\frac{(n+1)!(2n)!}{n!(2n+2)!}.\tag{2}$$
The above expression simplifies to
$$\frac{1}{2(2n+1)},$$
which has limit $0$ as $n\to\infty$. Thus the series converges. Fast.
We do the Ratio Test on the series as follows.
$$\lim_{n \to \infty}\left|\frac{\frac{(n + 1)!}{(2n + 2)!}}{\frac{n!}{(2n)!}}\right|$$
$$= \lim_{n \to \infty}\left|\frac{(n + 1)!}{(2n + 2)!} \times \frac{(2n)!}{n!}\right|$$
$$= \lim_{n \to \infty}\left|\frac{n + 1}{(2n + 2)(2n + 1)}\right|$$
$$= \lim_{n \to \infty}\left|\frac{1}{2(2n + 1)}\right|$$
$$= 0.$$

Since this limit is $0 &lt; 1,$ the series must converge.