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