Answer
Converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}-(\ln n)^{4}}$$
Compare with $\sum \frac{1}{n^{3/2}}$, which is a convergent $p-$series. Using the limit comparison test gives:
\begin{align*}
\lim _{n \rightarrow \infty}\frac{a_n}{b_n}&=\lim _{n \rightarrow \infty} \frac{n^{3 / 2}-(\ln n)^{4}}{n^{3 / 2}}\\
&=1-\lim _{n \rightarrow \infty} \frac{ (\ln n)^{4}}{n^{3 / 2}} \\
&= 1
\end{align*}
Thus, $\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}-(\ln n)^{4}} $ also converges.
