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}\\
&=\lim _{n \rightarrow \infty} \frac{n^3/n^3-n^2/n^3}{n^{3}/n^3+1/n^3}\\
&=\lim _{n \rightarrow \infty} \frac{1-1/n }{1+1/n^3}\\
&=1
\end{align*}
Since $\displaystyle \sum_{n=1}^{\infty} \frac{ 1}{n^2}$ is convergent ($p-$ series $p>1$) , then $\displaystyle\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+1}$ is also convergent.