Testing convergence of $\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$ Can anyone help me to prove whether this series is convergent or divergent:
$$\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$$
I tried using the ratio test, but the limit of the ratio in this case is equal to 1 which is inconclusive in this case. Any hints please!

Hint: You cannot use the alternating series test. Since 
$$ \lim_{n\to \infty}\frac{4^n n!^2}{(2n)!} \neq 0. $$ You can use the identity
$$ (2n)!=\Gamma(2n+1)={\frac {n  {4}^{n}\Gamma  \left( n \right) \Gamma 
 \left( \frac{1}{2}+n \right) }{\sqrt {\pi }}},$$
or Stirling approximation to see this.
