Answer
Divergent
Work Step by Step
The Comparison Test states that the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ is convergent if $p\gt 1$ and divergent if $p\leq 1$.
$a_{n}-\frac{1}{n}=\frac{n^{2}+1}{n^{3}+1}-\frac{1}{n}$
$=\frac{(n^{2}+1)n-(n^{3}+1)}{(n^{3}+1)n}$
$=\frac{n-1}{(n^{3}+1)}\geq 0$ for all $n$
Since, $\sum_{n=1}^{\infty}\frac{1}{n}$ is divergent, $\sum_{n=1}^{\infty}\frac{n^{2}+1}{n^{3}+1}$ is also divergent.
Hence, the series $\sum_{n=1}^{\infty}\frac{n^{2}+1}{n^{3}+1}$ is divergent.
