[College Calculus] Simplifying a ratio of factorials

The question is to prove that the following infinite series converges for all x:

Σ(n=0, Infinity) of ((-1)^n *x^2n ) / (2^2n * (n!)^2)

Sorry I couldn't format that any better, it's messy to try and type it. I applied the ratio test and I've ended up with the expression -x(n!)^2 / 2((n+1)!)^2

What I need help with is firstly, have I gone astray already? And secondly, if I'm right so far, how can I simplify the factorial terms when they're being squared like this?