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