Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{n!}{ (2n+1)!}$
Apply Ratio Test.
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)!}{ (2(n+1)+1)!}}{\dfrac{n!}{ (2n+1)!}}|$
$\implies \lim\limits_{n \to \infty}|(\dfrac{n+1}{(2n+2)(2n+3)})|=\lim\limits_{n \to \infty}|(\dfrac{n+1}{4n^2+6n+4n+6})|$
Thus, $\lim\limits_{n \to \infty}|(\dfrac{\dfrac{1}{n}+\dfrac{1}{n^2}}{4+\dfrac{10}{n}+\dfrac{6}{n^2}})|=0 \lt 1$
Thus, the series converges by the ratio test.