TITLE: Is the series $\sum_\limits{n=1}^\infty\frac{n}{n^3+1}$ convergent or divergent?
QUESTION [7 upvotes]: Is the series $$\sum_\limits{n=1}^\infty\frac{n}{n^3+1}$$ convergent or divergent?
My answer is the following. Could anyone tell me if it is correct?
Since $$0<\frac{n}{n^3+1}<\frac{1}{n^2}\;\;,\;\;\;\;\forall n\in\mathbb{N}$$ and the series $$\sum_\limits{n=1}^\infty\frac{1}{n^2}$$ is convergent,
by applying comparison test, we get that the series $$\sum_\limits{n=1}^\infty\frac{n}{n^3+1}$$ is convergent too.

REPLY [3 votes]: Let me add also some simple way in such situations: if we have series with positive general members $a_n, b_n$ and $a_n \sim b_n$ when $n \to \infty$, then series converges and diverges simultaneously. In your case
$$\frac{n}{n^3+1} \sim \frac{1}{n^2}$$