Answer
Converges
Work Step by Step
Given $$ \sum_{n=2}^{\infty} \frac{1}{n^2(\ln n)^{3}}$$
Compare with convergent series $ \sum_{n=2}^{\infty} \frac{1}{n^2 }$ ($p-$series $p>1$) , then by using the limit comparison test, we get:
\begin{align*}
\lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty} \frac{n^2}{n^2(\ln n)^{3}}\\
&= \lim_{n\to \infty} \frac{1}{ (\ln n)^{3}}\\
&=0
\end{align*}
Thus the given series also converges.
