Answer
Diverges
Work Step by Step
Given $$ \sum_{n=2}^{\infty} \frac{n^{3}-2 n^{2}+n-4}{2 n^{4}+3 n^{3}-4 n^{2}-1}$$
Compare with the divergent series $ \displaystyle\sum_{n=2}^{\infty} \frac{1}{2 n }$; then by using the limit comparison test
\begin{align*}
\lim_{n\to \infty } \frac{a_n}{b_n}&= \lim_{n\to \infty } \frac{n^{4}-2 n^{3}+n^2-4n}{2 n^{4}+3 n^{3}-4 n^{2}-1}\\
&= \lim_{n\to \infty } \frac{1-2/n+1/n^2-4/n}{2+3/n-4 /n^{2}-1/n^4}\\
&=\frac{1}{2}\gt 0
\end{align*}
Thus the given series also diverges.
