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