Timeline for Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 10, 2023 at 5:39 | comment | added | Fedor Petrov | @IosifPinelis A bit more general series $\sum \frac{\cos a n \sin(an+b)}n$ (in our situation $a=b=1$) is convergent if and only if $b/\pi \notin \mathbb{Z}$. | |
| Nov 9, 2023 at 22:12 | history | closed |
user99863 Matthias Ludewig Christian Remling Boris Bukh GH from MO |
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| Nov 9, 2023 at 21:44 | comment | added | Iosif Pinelis | @FedorPetrov : What do you mean by "only $1/\pi\notin\mathbb Z$ [is needed]"? This just follows by Dirichlet's test. | |
| Nov 9, 2023 at 21:26 | review | Close votes | |||
| Nov 9, 2023 at 22:15 | |||||
| Nov 9, 2023 at 21:19 | comment | added | Fedor Petrov | @Aleksei Irrationality of $\pi$ is not needed, by the way, only $1/\pi \notin \mathbb{Z} $ | |
| Nov 9, 2023 at 21:11 | answer | added | Carlo Beenakker | timeline score: 2 | |
| Nov 9, 2023 at 20:57 | comment | added | user516435 | Thanks, I forgot my trigonomical formulas. | |
| Nov 9, 2023 at 19:14 | comment | added | Robert Israel | Indeed, $\cos(n) \sin(n+1) = \frac{\sin(2n+1)}{2} + \frac{\sin(1)}{2}$. The sum of a convergent series and a divergent series is divergent. | |
| Nov 9, 2023 at 18:55 | comment | added | Aleksei Kulikov | Average value of $\cos(x)\sin(x+1)$ is $\sin(1)/2 \approx 0.42\ldots \neq 0$, so the sum should diverge. I think it wouldn't be too hard to prove using some sort of equidistribution (and, of course, irrationality of $\pi$), but I'll leave it to someone else. | |
| S Nov 9, 2023 at 18:48 | review | First questions | |||
| Nov 9, 2023 at 21:15 | |||||
| S Nov 9, 2023 at 18:48 | history | asked | user516435 | CC BY-SA 4.0 |