Answer
Divergent
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n^{2}-5 n}{n^{3}+n+1}$$
Use the Limit Comparison Test with $a_n =\dfrac{n^{2}-5 n}{n^{3}+n+1}$ and $b_n=\dfrac{1}{n}$
\begin{align*}
\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&=\lim _{n \rightarrow \infty} \frac{n^{3}-5 n^2}{n^{3}+n+1}\\
&=\lim _{n \rightarrow \infty} \frac{n^{3}/n^3-5 n^2/n^3}{n^{3}/n^3+n/n^3+1/n^3}\\
&=1
\end{align*}
Since $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$ is divergent , then $\displaystyle\sum_{n=1}^{\infty} \frac{n^{2}-5 n}{n^{3}+n+1}$ is also divergent.