Answer
Divergent
Work Step by Step
Let us consider $a_n=\lim\limits_{n \to \infty} \dfrac{\sqrt n}{\ln n}$
Here, $\lim\limits_{n \to \infty} \dfrac{\sqrt n}{\ln (n)}=\dfrac{\infty}{\infty}$
Thus, the limit has an indeterminate form so we will take the help of L-Hospital's rule.
Now, $\lim\limits_{n \to \infty} \dfrac{\dfrac{1}{2 \sqrt n}}{\dfrac{1}{n}} =\lim\limits_{n \to \infty} \dfrac{(\sqrt n)}{(2)}=\infty $
Thus, the series is divergent.