Convergence of Series $\sum 4^n(n!)^2/(2n)!$ [duplicate]

Consider the formal series $$\sum_{n=1}^\infty \frac{(n!)^2}{(2n)!}4^n.$$

Is there a way to test for the convergence/divergence of this series without explicitly using Stirling's approximation?

We have $$\lim_{n\to\infty} \frac{(n!)^2}{(2n)!}4^n=\lim_{n\to\infty}4^n\frac{n}{2n}\frac{n-1}{2n-1}\cdots\frac{2}{n+2}\frac{1}{n+1}.$$

What is the limit? And if the limit turned out to be 0, what can we do?