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

Determine the series converges absolutely or conditionally.


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

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


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.


marked as duplicate by Fabian, Robert Z real-analysis Jul 30 '18 at 10:21

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  • $\begingroup$ Do you know the integral test? $\endgroup$ – Fabian Jul 30 '18 at 10:14

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.


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