Timeline for "Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2022 at 23:10 | review | Close votes | |||
| Jul 16, 2022 at 18:30 | |||||
| Jul 11, 2022 at 11:21 | answer | added | Iosif Pinelis | timeline score: 13 | |
| Jul 10, 2022 at 21:15 | vote | accept | Paresseux Nguyen | ||
| Jul 10, 2022 at 7:20 | answer | added | Carlo Beenakker | timeline score: 12 | |
| Jul 10, 2022 at 3:12 | comment | added | Alapan Das | Assuming that the limit exists, $\lim_{n \to \infty} \frac{x_n}{n^a}=c$, for some constant and estimating $S_n=\sum_{k=1}^{n} x_k=c\int_{1}^{n} x^a dx +\epsilon_n$, we get from the recurrence relation, $c[(n+1)^a-n^a]=\frac{n}{S_n}$, for $n>>0$, $cn^{a-1}=\frac{a+1}{c}n^{-a} \Rightarrow a=\frac{1}{2}$ and $c=\sqrt{3}$. | |
| Jul 10, 2022 at 2:03 | comment | added | Paresseux Nguyen | @WillSawin: Right. | |
| Jul 10, 2022 at 1:41 | comment | added | Will Sawin | Given that this comes from a contest-problem-related source and then seems contest-problem-y, I would guess that where this problem comes from is very similar to where contest problems come from. But then I realized that I don't really know how contest problems are usually written. | |
| Jul 10, 2022 at 1:14 | comment | added | Steven Landsburg | @NoamD.Elkies: That thread establishes that there exists a person who does not know how to determine whether the limit exists. This seems to be necessary, but not sufficient, for the question to be open. | |
| Jul 9, 2022 at 23:37 | comment | added | Noam D. Elkies | The open question seems to be proving that the limit exists (which is needed for the validity of the derivation of the limit's value). | |
| Jul 9, 2022 at 23:36 | comment | added | Will Jagy | @StevenLandsburg one comment links to link.springer.com/book/10.1007/978-3-030-77139-3 and says this problem is a special case of something in a section of open problems numbered 1.30. As the book is about "ingenious computations" in contests, I suspect this problem can be solved some clever way | |
| Jul 9, 2022 at 23:25 | comment | added | Steven Landsburg | There is nothing in the thread you linked to that suggests this problem is open. | |
| Jul 9, 2022 at 22:58 | history | asked | Paresseux Nguyen | CC BY-SA 4.0 |