Determine whether the series converges $$\sum_{k=1}^\infty \frac{(k!)^2}{(2k)!}$$
Attempt: I used ratio test, but I guess I am making a mistake in cancelling out terms.$$\lim_{k\rightarrow \infty}\frac{((k+1)!)^2}{(2(k+1))!} \frac{(2k)!}{(k!)^2}=\lim_{k\rightarrow \infty } \frac{k+1}{2}$$
I am not experienced with factorials. For example, I know that $(k+1)!=k!(k+1)$, but I cant figure what $(2(k+1))!$ equals to. Help appreciated.