Answer
Converges
Work Step by Step
Consider $a_n=\sin^n (\dfrac{1}{\sqrt n})$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\sin^n (\dfrac{1}{\sqrt n})|}=\sin \lim\limits_{n \to \infty} (\dfrac{1}{\sqrt n})$
or, $\sin (\dfrac{1}{\infty})=\sin 0=0$
so, $l \lt 1$
Thus, the given series Converges by the Root Test.