Answer
Diverges
Work Step by Step
Let us consider $a_n=\dfrac{3n!}{n!(n+1)(n+2)!}$
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{3(n+1)!}{(n+1)!(n+2)!(n+3!}}{\dfrac{3n!}{n!(n+1)!(n+2)!}}|$
$\implies \lim\limits_{n \to \infty}|\dfrac{(3n+3)(3n+2)(3n+1)}{(n+1)(n+2)(n+3)}|=\lim\limits_{n \to \infty}|\dfrac{(3+\dfrac{3}{n})(3+\dfrac{2}{n})(3+\dfrac{1}{n})}{(1+\dfrac{1}{n})(1+\dfrac{2}{n})(1+\dfrac{3}{n})}|=\dfrac{3 \cdot 3 \cdot 3}{ 1 \cdot 1 \cdot 1}=27 \gt 1$
Thus, the series diverges by the ratio test.