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