Convergence of a power series at points where ratio test is inconclusive

Question

I need to find the interval of convergence of the following power series using either the ratio test , integral test or comparison test. Using the ratio test I found that it will converge for $ -4 < x < 4 $ but it proves inconclusive at $ x = \pm 4 $
$$ \sum_{n=0}^{\infty} \frac{(n!)^2 x^n}{(2n)!} $$

Answer 1

When $x=\pm 4$, $$ |\frac{(n!)^2}{(2n)!} x^n|=\frac{n! 2^n }{1\cdot 3\cdot \cdots (2n-1)}=\frac{2\cdot 4\cdot \cdots (2n)}{1\cdot 3\cdot \cdots (2n-1)} $$ This does not converge to zero, so the series is divergent at both points.