Answer
convergent
Work Step by Step
Given
$$ \sum_{n=1}^{\infty} \frac{n-1}{n^{3}+1}$$
Use the Limit Comparison Test with $a_{n}=\dfrac{n-1}{n^{3}+1}$ and $b_{n}=\dfrac{1}{ n^2}$
\begin{align*}
\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&=\lim _{n \rightarrow \infty} \frac{ n^3-n^2 }{ n^3+1}\\
&=1
\end{align*}
since $\displaystyle\sum_{n=1}^{\infty} \frac{1}{ n^2}$ is convergent $(p-\text { series } p>1),$ then $\displaystyle\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+1}$ also convergent