calculus - Convergence of $sumfrac{|cos n|}{nlog n}$

I wonder if the series $$\sum_{n=1}^\infty\frac{|\cos n|}{n\log n}$$ converges.

I tried to applying the condensation test, getting $$\sum\frac{2^n|\cos 2^n|}{2^n\log{2^n}}=\sum\frac{|\cos 2^n|}{n\log 2}$$ but I don't know how to show it converges?

Am I in the right way?

Answer

Note that

$$|\cos n| \geqslant \cos^2 n = \frac1{2} + \frac1{2}\cos(2n).$$

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.