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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!

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  • $\begingroup$ This is the line between conditional convergence and absolute convergence. In this case, it converges conditionally. $\endgroup$ – Simply Beautiful Art Dec 3 '16 at 18:56
  • $\begingroup$ Ah, I see what you're saying. So just to confirm, if the series of absolute values diverges, this just tells me that it is not absolutely convergent? So the original series doesn't necessarily diverge, it could either diverge of converge conditionally? $\endgroup$ – melm Dec 3 '16 at 18:59
  • $\begingroup$ Yes. That is what is happening. As a last remark, when some convergence tests fail, it doesn't necessarily mean something is divergent. Likewise, some tests, like the term test, cannot tell us if something converges, but it can tell us if it diverges. $\endgroup$ – Simply Beautiful Art Dec 3 '16 at 19:01
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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.

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