Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{n!}{ (2n+1)!}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)!}{ (2(n+1)+1)!}}{\dfrac{n!}{ (2n+1)!}}|$
Thus, we have $l=\lim\limits_{n \to \infty}|(\dfrac{n+1}{(2n+2)(2n+3)})|=\lim\limits_{n \to \infty}|(\dfrac{n+1}{4n^2+6n+4n+6})|=\lim\limits_{n \to \infty}|(\dfrac{1/n+1/n^2}{4+10/n+6/n^2})|$
So, $l=0 \lt 1$
Hence, the series Converges by the ratio test.