Answer
Converges to $1$
Work Step by Step
Consider $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \sqrt n \sin (\dfrac{1}{\sqrt n})$
But $\lim\limits_{n \to \infty} \sqrt n \sin (\dfrac{1}{\sqrt n})=\dfrac{0}{0}$
Need to apply L-Hospital's rule.
So, $ \lim\limits_{n \to \infty} \dfrac{\cos (1/\sqrt n) \cdot (-1/2n^{3/2})}{\dfrac{-1}{2n^{3/2}}}=\lim\limits_{n \to \infty} \cos (\dfrac{1}{\sqrt n})$
or, $=1$
Hence, $\lim\limits_{n \to \infty} a_n=1$ and {$a_n$} is Convergent and converges to $1$
