I wonder if the series ∞∑n=1|cosn|nlogn converges.
I tried to applying the condensation test, getting ∑2n|cos2n|2nlog2n=∑|cos2n|nlog2 but I don't know how to show it converges?
Am I in the right way?
Answer
Note that
|cosn|⩾
Hence
\sum_{k=2}^n \frac{|\cos k|}{k \log k}\geqslant \frac1{2}\sum_{k=2}^n \frac{1}{k \log k}+\frac1{2}\sum_{k=2}^n \frac{\cos(2k)}{k \log k}.
The series on the LHS diverges because the first sum on the RHS diverges by the integral test and the second sum converges by Dirichlet's test.
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