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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

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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. –  Alex Becker Mar 23 '11 at 1:55
On the other hand, the positive and negative terms "should" cancel each other out for the most part. I expect convergence. –  Michael Lugo Mar 23 '11 at 1:59
I expect it to converge... Probably try a variation of generalized alternate test (Dirichlet test)? –  user17762 Mar 23 '11 at 2:01

1 Answer 1

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))".

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Thanks Jonas. –  anonymous Mar 23 '11 at 2:02
And just for the record: the answer is that the series converges. –  Hans Lundmark Mar 23 '11 at 7:59

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