Answer
Converges
Work Step by Step
Here, we have $L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(2^{n+1})((n+1)!)((n+1)!)}{(2(n+1))!}}{\dfrac{(2^n)(n!)(n!)}{(2n)!}}|$
$\implies L=(2) \lim\limits_{n \to \infty}|\dfrac{(n+1)^2}{4n^2+2n+4n+2}|= \lim\limits_{n \to \infty}|\dfrac{1+\dfrac{1}{n^2}+\dfrac{2}{n}}{4+\dfrac{6}{n}+\dfrac{2}{n^2}}| \cdot (2)=\dfrac{1}{2} \lt 1$
Thus, the series converges by the ratio test.