Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{n+2}{3^n}$
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.
Then $L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n+1}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)+2}{3^{n+1}}}{\dfrac{n+2}{3^n}}|$
$\implies \lim\limits_{n \to \infty}|\dfrac{n+3}{3(n+2)}|=\lim\limits_{n \to \infty}|\dfrac{n+3}{3n+6}|$
and $L=\lim\limits_{n \to \infty}|\dfrac{1+3/n}{3+6/n}|=\dfrac{1}{3} \lt 1$
Hence, the series Converges by the ratio test.