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?