Answer
(c)
Work Step by Step
To show that the series $\sum_{n=1}^\infty \frac{n}{n^3+1}$ converges by the Direct Comparison Test, we consider $a_n=\frac{n}{n^3+1}$ and then we need to find a convergent series $\sum_{n=1}^\infty b_n$ satisfying $a_n\leq b_n$ for all $n\geq 1$.
It means that the inequality (a) is not satisfied and the inequality (b) is also not satisfied because $\sum_{n=1}^\infty \frac{1}{n}$ is divergent.
Thus, the inequality that can be used is (c) $\frac{n}{n^3+1}\leq \frac{1}{n^2}$ since $\sum_{n=1}^\infty \frac{1}{n^2}$ is a convergent $p-$series with $p=2$.
