Timeline for Series whose convergence is not known
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 30, 2017 at 0:19 | history | edited | Robert Israel | CC BY-SA 3.0 |
added 73 characters in body
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| Sep 29, 2017 at 18:33 | comment | added | Sam Hopkins | See: mathoverflow.net/questions/282259/…. | |
| Oct 19, 2016 at 18:19 | comment | added | Robert Israel | If you're just saying that the series appears to converge, then I agree: I would be very surprised if it did not converge. But that's not a proof. | |
| Oct 19, 2016 at 7:31 | comment | added | Robert Israel | I'm not convinced by what's at that link. Since $\sum_n |\sin(nt\pi)|^n/n$ diverges for $t$ in a dense $G_\delta$, and in particular for uncountably many irrational $t$, it seems to me you need to use more than just Weyl equidistribution. | |
| Oct 18, 2016 at 21:07 | comment | added | Jack D'Aurizio | Actually, by combining Abel's summation formula with integral estimates, it looks that $\sum_{n\geq 1}\frac{\left|\sin n\right|^n}{n}$ is convergent: math.stackexchange.com/questions/823816/… The MSE user Kirill also started some research about efficient tailor-made numerical methods for the computation of such "pseudorandom" series. | |
| Jun 16, 2014 at 14:55 | comment | added | user23855 | Sorry for bumping, but do you have a reference for the first series? There's a bounty on it here math.stackexchange.com/questions/823816/… | |
| Mar 11, 2012 at 18:13 | comment | added | Robert Israel | If $t = p/(2q)$ where $p$ and $q$ are odd integers, then $|\sin(nt\pi)|=1$ whenever $n$ is an odd multiple of $q$, and so the series diverges for such $t$. Such $t$ form a dense subset of $\mathbb R$. The set where a series of nonnegative continuous functions diverges is a $G_\delta$, so we have a dense $G_\delta$. For the convergence a.e., note that $$ \int_0^1 \frac{|\sin(nt\pi)|^n}{n}\ dt \sim \frac{C}{n^{3/2}}$$ so the series converges in $L^1[0,1]$, therefore the sum is finite almost everywhere in $[0,1]$ (and, by periodicity, in $\mathbb R$). | |
| Mar 8, 2012 at 2:50 | comment | added | Portland | @Robert, this is very nice. Do you have a reference for the second series? | |
| May 25, 2011 at 17:21 | history | edited | Robert Israel | CC BY-SA 3.0 |
removed extra |
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| May 24, 2011 at 17:01 | history | answered | Robert Israel | CC BY-SA 3.0 |