Answer
Converges
Work Step by Step
Given $$\sum_{n=2}^{\infty} \frac{n^{2}+1}{n^{3.5}-2}$$
Compare with the convergent series $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n^{1.5}}$ ($p-$series $p>1$), by using the Limit Comparison Test
\begin{align*}
\lim_{n\to\infty} \frac{a_n}{b_n} &=\lim_{n\to\infty} \frac{n^{3.5}+n^{1.5}}{n^{3.5}-2}\\
&=1
\end{align*}
Thus the given series also converges.