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Given the following series:

$$\sum_{n=1}^{+\infty} \frac{|\cos n|^n}{n}$$

(being a series with non-negative terms), does it converge or diverge?

Unfortunately, I can't prove in any way or convergence or divergence of this series. Any ideas?

Daniel Fischer
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πρόσεχε
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  • With $7000$ terms it's ( numerically ) $\approx 1.6\color{#f00}{666683043389468266458468225848077445946578253126}$. It's slightly larger than ${\pi^{2} \over 6} \approx 1.6449340668482264364724151666460251892189499012068$. Maybe, a nice question is to prove that the series is $> {\pi^{2} \over 6}$. – Felix Marin Oct 19 '16 at 04:21

1 Answers1

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The series' convergence is famously unknown. Specifically, see this Mathoverflow post about the similar series with $\sin n$ instead of $\cos n$:

For an interesting example, take $\sum_{n=1}^\infty \frac{|\sin(n)|^n}{n}$. Deciding whether or not this converges seems to require more knowledge than is currently available about the rational approximations of $\pi$. The series $\sum_{n=1}^\infty \frac{|\sin(n t \pi)|^n}{n}$ converges for almost every real $t$ (in the sense of Lebesgue measure), but diverges for $t$ in a dense $G_\delta$ subset of $\mathbb R$.

Deciding whether $\sum_{n=1}^\infty \frac{|\cos(n)|^n}{n}$ converges, you will run into all of the same problems as with $\sum_{n=1}^\infty \frac{|\sin(n)|^n}{n}$.

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Caleb Stanford
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  • Actually, $\sum_{n\geq 1}\frac{|\sin n|^n}{n}$ is convergent (http://math.stackexchange.com/questions/823816/is-sum-limits-n-1-infty-frac-sin-nnn-convergent/836054#836054) and the linked answer also spawned some research for efficient methods for its numerical computation. – Jack D'Aurizio Oct 18 '16 at 20:58
  • @JackD'Aurizio Holy shit, so you solved an open series problem in a math stackexchange post? Then you should comment on Robert Israel's mathoverflow post...and do you have a link to a paper where this has been peer-reviewed? – Caleb Stanford Oct 18 '16 at 21:00
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    actually I did not know it was an open problem, when I solved it here, so the only peer review actually made is by the MSE community. Anyway, I guess I will notify Robert about that, thanks for the suggestion. – Jack D'Aurizio Oct 18 '16 at 21:04