Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{n+2}{3^n}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n+1}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)+2}{3^{n+1}}}{\dfrac{n+2}{3^n}}|$
Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{n+3}{3(n+2)}|=\lim\limits_{n \to \infty}|\dfrac{n+3}{3n+6}|=\dfrac{1}{3} \lt 1$
Hence, the series Converges absolutely by the ratio test.