Convergence of absolute value of a series I have the following series:
$$\sum_{n=1}^\infty \frac{n(-1)^n}{n^2+1} \\$$
If I use the alternating series test, I find that it is a convergent series.
My question is, if I use the theorem: "A series is convergent if the series of absolute values is convergent" I find that the series of absolute values diverges, which doesn't make sense as by the alternating series test the original series converges. 
In this sense, can I conclude that the original series also diverges if the series of absolute values diverges? Or does the statement only work one way.
Thanks!

The implication only works one way. A series is said to be absolutely convergent if the series of absolute values converges-and this implies that it is convergent. A series is said to be conditionally convergent if the series of absolute values diverges but the series itself converges. What you have is a conditionally convergent series.
