Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{8}{(3+\dfrac{1}{n})^{2n}}$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\dfrac{8}{(3+\dfrac{1}{n})^{2n}}|}=\lim\limits_{n \to \infty} \dfrac{(8)^{1/n}}{(3+\dfrac{1}{n})^{2}}$
or, $=\dfrac{1}{9}$
so, $l \lt 1$
Thus, the given series Converges by the Root Test.