Does the series $\sum_{n=2}^\infty \frac{(-1)^n}{n \ln(n)}$ converge or diverge?

Question

I'm a student, solving convergence problems today.

I can show that the series $\sum_{n=2}^\infty \frac{1}{n \ln{n}}$ diverges by integral test ($F(x) = \ln{(\ln{x})}, F'(x) = f(x) = \frac{1}{x \ln{x}}$)

But I don't have any idea for the convergence of $\sum_{n=2}^\infty \frac{(-1)^n}{n \ln(n)}$

any help would be appreciated.

Answer 1

Yes, this does converge. In alternating series', as long as the absolute value of the numbers get closer to zero, it converges. 1/ln(n)n always get closer to 0 as n gets closer to infinity.

Answer 2

The alternating series test immediately implies that the series converges.