Testing convergence of $\sum\limits_{n=2}^{\infty} \frac{\cos{\log{n}}}{n \cdot \log{n}}$

Question

Does the series: $$\sum\limits_{n=2}^{\infty} \frac{\cos(\log{n})}{n \cdot \log{n}}$$ converge or diverge?

I know that $|\cos(\log{n})| \leq 1$, but i really cannot apply it here. Any ideas on how to attack this problem

A quick try using $|\cos(\log{n})| \leq 1$ and the integral test gives that the $n$th partial sum is $\leq log(log(n))$, so if this diverges it does so very slowly.
On the other hand, the positive and negative terms "should" cancel each other out for the most part. I expect convergence.
I expect it to converge... Probably try a variation of generalized alternate test (Dirichlet test)?

Jonas Meyer · Accepted Answer · 2011-03-23 02:01:35Z

up vote 9 down vote accepted

This problem appears in the Nordic university-level mathematics team-competition, NMC, 2010, with solution at the beginning of the following pdf: http://cc.oulu.fi/~phasto/competition/2010/solutions2010.pdf.

The search was series "cos(log(n))".

answered Mar 23 '11 at 2:01

Jonas Meyer
35k161129

Thanks Jonas. – anonymous Mar 23 '11 at 2:02

2

And just for the record: the answer is that the series converges. – Hans Lundmark Mar 23 '11 at 7:59

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