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. — Alex Becker, Mar 23 '11 at 01:55
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 01:59
I expect it to converge... Probably try a variation of generalized alternate test (Dirichlet test)? — , Mar 23 '11 at 02:01
A numerical value of the sum is in https://oeis.org/A363677 . — R. J. Mathar, Jun 14 '23 at 16:37

score 12 · Accepted Answer · answered Mar 23 '11 at 02:01

12

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 02:01

Jonas Meyer

4

And just for the record: the answer is that the series converges. – Hans Lundmark Mar 23 '11 at 07:59

score 2 · Answer 2 · answered Jun 29 '17 at 14:59

2

In case the link Jonas gave would be broken in the future, here is the solution:

answered Jun 29 '17 at 14:59

2 Answers2