Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{2^{n+1}}{n3^{n-1}}$
In order to solve the given series we will take the help of Ratio Test. This test states that when the limit $L \lt 1$ , the series converges and for $L \gt 1$, the series diverges.
$L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{2^{(n+2)}}{(n+1)3^{n}}}{\dfrac{2^{(n+1)}}{n3^{(n-1)}}}|$
$\implies \lim\limits_{n \to \infty}|\dfrac{2n}{3(n+1)}|=\lim\limits_{n \to \infty}|\dfrac{2n}{3n+3}|$
and $L=\lim\limits_{n \to \infty}|\dfrac{2}{3+3/n}|=\dfrac{2}{3} \lt 1$
Hence, the series Converges by the ratio test.