Answer
Diverges
Work Step by Step
Let us consider $a_n=\dfrac{(n-1)!}{(n+1)^2}$
In order to solve the given series we will take the help of Ratio Test. This test states that when the limit $L \lt 1$ , the series converges and for $L \gt 1$, the series diverges.
Then $L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n)!}{(n+2)^2}}{\dfrac{(n-1)!}{(n+1)^2}}|$
$\implies \lim\limits_{n \to \infty}|\dfrac{n(n+1)^2}{(n+2)^2}|=\lim\limits_{n \to \infty}|\dfrac{n^3+n+2n^2}{n^2+4n+4}|$
or, $L=\lim\limits_{n \to \infty}|\dfrac{1+\dfrac{1}{n^2}+\dfrac{2}{n}}{\dfrac{1}{n}+\dfrac{4}{n^3}+\dfrac{4}{n^2}}|=\infty$
Hence, the series Diverges by the ratio test.