Consider the series
$$\sum_{n=1}^\infty\frac{(-1)^n}{n\log^2(n+1)}.$$
Determine whether it converges absolutely or conditionally.
My attempt
S=$\sum_{n=1}^{\infty}( -1)^n$ an
an 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.
Best Answer
For the absolute convergence by cauchy condensation test we can consider the convergenge of the condensed series $\sum 2^n a_{2^n}$ that is
$$\sum \frac{2^n}{2^n(\log^2(2^n+1))}=\sum \frac{1}{\log^2(2^n+1)}$$
which converges by limit comparison test with $\sum \frac1{n^2}$ indeed
$$\frac{1}{\log^2(2^n+1)}\sim\frac1{n^2\log^2 2}$$