Answer
Converges
Work Step by Step
We are given that $a_k=\dfrac{(k!)^2}{(2k!)}$
Ratio Test states that when $\Sigma a_k$ is an infinite series with positive terms and, then $r=\lim\limits_{k \to \infty}\dfrac{a_{k+1}}{a_k}$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{k \to \infty}\dfrac{(k+1)^{2} (k!)^2(2k)!}{(k!)^2 (2k+2)(2k+1)(2k)!} \\= \lim\limits_{k \to \infty} \dfrac{(k+1)^2}{(2k+2)(2k+1)} \\=\dfrac{1}{4}$
Therefore, the series converges by the ratio test.
