Does this alternating series converge?

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?

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

• Plenty of series never get large, such as $1-1+1-1+\cdots,$ but still diverge. – zhw. Oct 29 '15 at 0:27