Answer
Diverges
Work Step by Step
Let $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln (\ln n)}$
But, $\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln (\ln n)}=\dfrac{\infty}{\infty}$
Apply L-hospital's rule such that $\lim\limits_{x \to k}\dfrac{p(x)}{q(x)}=\lim\limits_{x \to k}\dfrac{p'(x)}{q'(x)}$
$\implies \lim\limits_{n \to \infty}\dfrac{n \ln (n)}{n}=\infty$
Hence, the series diverges.
