I'm trying to prove the convergence of the following series: $$\sum_{n=2}^\infty \frac{(-1)^{n+1}}{n\ln n}$$

I started by applying the alternating series test, and calculated $$\left| \frac{(-1)^{n+1}}{n\ln n} \right| = \frac{1}{n\ln n} \to 0.$$

So, the alternating series must converge.

But the series $\sum_{n=2}^\infty \frac{1}{n\ln n}$ doesn't converge. So why does the alternate series converge?

  • $\begingroup$ Because the alternating series test applies. $\endgroup$ – zhw. Oct 29 '15 at 0:20
  • $\begingroup$ I thought that if an alternate serie converges absolutely $\implies$ the serie not alternate converges. $\endgroup$ – hlapointe Oct 29 '15 at 0:21
  • $\begingroup$ If any series converges absolutely then the series converges. But if $\sum |a_n|=\infty,$ it does not imply $\sum a_n$ diverges. $\endgroup$ – zhw. Oct 29 '15 at 0:23
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    $\begingroup$ The series is not absolutely convergent and is instead conditionally convergent. Many alternating series have this property, such as the alternating harmonic series. $\endgroup$ – Jeevan Devaranjan Oct 29 '15 at 0:23
  • $\begingroup$ @JeevanDevaranjan Ok. Now, I understand. Thanks for the tips. $\endgroup$ – hlapointe Oct 29 '15 at 0:26

The logical answer: Because the series is alternating, you can apply the test for alternate series. The other series is not alternating and thus the same argument does not apply.

The intuitive answer: The problem with convergence in this case is whether the infinite sum reaches infinity. The non-alternating series gets to big, but the minuses in the alternating series makes sure that the series does not grow like crazy.

Remember that a convergent series is not necessarily absolutely convergent. This only holds the other way around.

  • $\begingroup$ Plenty of series never get large, such as $1-1+1-1+\cdots,$ but still diverge. $\endgroup$ – zhw. Oct 29 '15 at 0:27
  • $\begingroup$ @zhw: True. But for positive series (or series with a tail behaving this way), getting "big" is the problem you want to avoid. $\endgroup$ – William Kurdahl Oct 29 '15 at 15:47

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