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