Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{(n+3)!}{(3!) (n!) (3^{(n)})}$
Apply ratio test.
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{((n+1)+3)!}{3! (n+1)! (3^{n+1})}}{\dfrac{(n+3)!}{(3!) (n!) (3^n)}}|$
$\implies (\dfrac{1}{3})\lim\limits_{n \to \infty}(\dfrac{n+4}{n+1})=(\dfrac{1}{3})\lim\limits_{n \to \infty}(\dfrac{1+4/n}{1+1/n})=(\dfrac{1}{3})(1)$
and $\dfrac{1}{3} \lt 1$
Thus, the series converges by the ratio test.
