The series $\sum_{1}^{\infty}\frac{2k!}{(k!)^2}$ = 4 diverges by the ratio test.
$\sum_{1}^{\infty}\frac{2k!}{(k!)^2} = (2) \sum_{1}^{\infty}\frac{k!}{(k!)^2} = (2)\sum_{1}^{\infty}\frac{1}{(k!)}$
Is this very bad algebra? When I compute the series $(2)\sum_{1}^{\infty}\frac{1}{k!}$ = 0 converges by the ratio test.