Answer
Diverges
Work Step by Step
Consider $a_n=\dfrac{3n!}{n!(n+1)(n+2)!}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{3(n+1)!}{(n+1)!(n+2)!(n+3)!}}{\dfrac{3n!}{n!(n+1)!(n+2)!}}|$
Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{(3n+3)(3n+2)(3n+1)}{(n+1)(n+2)(n+3)}|=\lim\limits_{n \to \infty}|\dfrac{(3+3/n)(3+2/n)(3+1/n)}{(1+1/n)(1+2/n)(1+3/n)}|$
So, $l=\dfrac{(3)(3)(3)}{1}=27 \gt 1$
Hence, the series Diverges by the ratio test.