Answer
Converges absolutely
Work Step by Step
Let us consider $u_n=n^2 (\dfrac{2}{3})^n$
Apply Ratio Test.
Here, we have $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n+1}} |=\lim\limits_{n \to \infty}|\dfrac{(n+1)^2 (\dfrac{2}{3})^{(n+1)}}{n^{(2)} (\dfrac{2}{3})^n}|$
or, $l=\dfrac{2}{3} \lt 1$
Hence, the series Converges absolutely by the ratio test.