Determine whether $\sum_{n=1}^{\infty}\frac{(-1)^n}{n \log^2(n+1)}$ converges absolutely or conditionally.

Question

Problem

Let $S = \sum_{n=1}^{\infty}\frac{(-1)^n}{n\log^2(n+1)}$.

Determine the series converges absolutely or conditionally.

Attempt

$S=\sum_{n=1}^{\infty}( -1)^n a_n$

$a_n$ is monotonically decreasing and it approaches zero when $n$ approaches infinity. So series is convergent .

Doubt

How to check for absolute convergence?

Ratio test fails here. Root test is of no use. I have attempted comparison tests using the fact that $n>\log(n)$, but no success there also.

Answer 1

One option is the condensation test, which says $\sum_{n\geq 1}\frac{1}{n(\log(n+1))^2}$ converges if and only if $\sum_{n\geq 0}\frac{2^n}{2^n(\log (2^{n+1}))^2}$ does.