Determine whether this series is absolutely convergent, conditionally convergent or divergent?

Question

The series $ \sum_{n=1}^{\infty} \frac{(-1)^n n}{n^2 + 1} $; is it absolutely convergent, conditionally convergent or divergent?

This question is meant to be worth quite a few marks so although I thought I had the answer using the comparison test, I think I'm supposed to incorporate the alternating series test.

Answer 1

Your series is convergent by Leibniz-theorem but not absolutely convergent as you can see by comparison with $\sum \frac{1}{n+1}$

Answer 2

The way @Fant walked is practical, but maybe this approach also helps:

Use the integral test. As $f(x)=\frac{x}{x^2+1}$ is positive monotonic decreasing function on $x\geq 2$, so the integral test then $\sum_2^{\infty}f(n)$ converges or diverges if $\int_2^{\infty}f(x)dx$ converges or diverges. But the integral is clearly diverges, so we have here what @Fant noted again.