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