Timeline for $\sum_{k=1}^{n} \arcsin(\sin(k))= a_n+b_n \pi$, for $a_n,b_n\in\mathbb{Z}$; what can be said about the sequence $(a_n,b_n )$?
Current License: CC BY-SA 4.0
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Jul 22 at 4:05 | history | bounty ended | CommunityBot | ||
| S Jul 22 at 4:05 | history | notice removed | CommunityBot | ||
| Jul 21 at 20:08 | comment | added | Srini | I have included lot more context now. If the number of my comments here grows, I will delete my comments in this question so that the bot doesn't move it to chat. Hopefully I can manage with the comments section in my answer or the answer body itself. | |
| Jul 21 at 18:14 | answer | added | Malo | timeline score: 3 | |
| Jul 21 at 14:48 | comment | added | Integrand | @Srini I think so, but I'm not sure. I would like to see a bit more elaboration in your answer before I award it the bounty, especially since you seem to have put a lot of thought into it. | |
| Jul 21 at 14:46 | comment | added | Srini | If $a_n =0$, is $b_n$ always $0$? If so, I can simplify the answer I posted | |
| Jul 20 at 21:54 | answer | added | Srini | timeline score: 3 | |
| S Jul 14 at 2:16 | history | bounty started | Integrand | ||
| S Jul 14 at 2:16 | history | notice added | Integrand | Draw attention | |
| Jul 13 at 16:27 | history | edited | Integrand | CC BY-SA 4.0 | added 60 characters in body |
| Jul 13 at 16:14 | history | edited | Integrand | CC BY-SA 4.0 | clarified definition |
| S Nov 21, 2023 at 22:07 | history | bounty ended | CommunityBot | ||
| S Nov 21, 2023 at 22:07 | history | notice removed | CommunityBot | ||
| Nov 14, 2023 at 3:13 | history | edited | Alex Ravsky | edited tags | |
| S Nov 13, 2023 at 20:49 | history | bounty started | Integrand | ||
| S Nov 13, 2023 at 20:49 | history | notice added | Integrand | Draw attention | |
| Dec 7, 2020 at 0:58 | history | edited | Integrand | CC BY-SA 4.0 | deleted 2 characters in body |
| Dec 6, 2020 at 20:00 | comment | added | richrow | Indeed, we can simply take $f(t)=\arcsin(\sin(2\pi t))$ and since $\int_{0}^{1}f(t)dt=0$ we have $(a_n+b_n\pi)/n\to 0$. It remains to prove that $a_n,b_n\sim n$ (up to constant). | |
| Dec 6, 2020 at 19:53 | comment | added | richrow | Probably this can be useful: sequence $(k/(2\pi))$ is equidistributed modulo 1 (en.wikipedia.org/wiki/Equidistribution_theorem) and it follows that for any Riemann-integrable function the sum $\frac{1}{N}\sum_{k=1}^{N}f(\{k/2\pi\})$ tends to the $\int_{0}^{1}f(t)dt$. I guess this can help to approximate this sum, but not sure. | |
| Dec 6, 2020 at 18:04 | history | edited | Integrand | CC BY-SA 4.0 | added 405 characters in body; edited tags |
| Dec 6, 2020 at 13:21 | answer | added | jjagmath | timeline score: 6 | |
| Dec 6, 2020 at 7:25 | answer | added | Claude Leibovici | timeline score: 3 | |
| Dec 5, 2020 at 19:42 | comment | added | Integrand | @StevenStadnicki, I'm using $-\pi/2 \le \arcsin(x)\le \pi/2$. So for instance, $\arcsin(\sin(10))=3\pi -10$ and $\arcsin(\sin(13))=13-4\pi$. In particular, it's not as you suggest because for $n=24$ the sum vanishes but your version would give $300-72\pi$. | |
| Dec 5, 2020 at 19:39 | comment | added | Steven Stadnicki | What range for arcsin are you using? Is this basically $\sum_k \left(k\bmod 2\pi\right)$ ? | |
| Dec 5, 2020 at 19:34 | history | edited | Integrand | CC BY-SA 4.0 | typo |
| Dec 5, 2020 at 17:33 | comment | added | dezdichado | have you tried restricting to the subsequence $\{b_n\neq 0\}$ and then feeding $\{b_n\}$ to the inverse sequence encyclopedia? I feel like any problem involving some sort of irrationality measure or convergents of $\pi$ either end up having a trivial recurrence or completely unsolved. Very nice question though. | |
| Dec 5, 2020 at 17:12 | history | edited | Integrand | CC BY-SA 4.0 | added 83 characters in body |
| Dec 5, 2020 at 17:07 | history | asked | Integrand | CC BY-SA 4.0 |