Talk:Riemann series theorem

Long on theory, short on specifics

Hi! I was just poking through some articles, and after looking at "absolute convergence", I wound up here. I didn't find a worked example of how a conditionally convergent series can actually be re-arranged to yield two different sums in either place. I think such an example would be helpful to the general reader. I also think this article should mention earlier results of Cauchy and Dirichlet. DavidCBryant 13:51, 23 January 2007 (UTC)

Could some clever soul please add a proof of this interesting theorem. I swear the one given in my book on analysis is just wrong! —Preceding unsigned comment added by 82.17.66.106 (talk) 19:13, 22 March 2008 (UTC)

Add ?

criticism for Riemann's theorem on a sum of conditionaly convergent series (pdf,eng,26KB) http://alexander-conon.narod.ru/science/crte.pdf

comment on "criticism for Riemann's theorem on a sum of conditionaly convergent series" (pdf,eng,26KB) http://alexander-conon.narod.ru/science/crte-c.pdf 82.151.112.75 (talk) 00:10, 25 June 2008 (UTC)

Is that correct?, nobody knows Alexander Conon. — Preceding unsigned comment added by 190.82.98.194 (talk) 14:04, 17 July 2013 (UTC)

A simpler example

Consider the conditionally convergent series

1 - 1 + 1/2 - 1/2 + 1/3 - 1/3 + -...

Permutation of this series taking p positives and q negatives has sum log(p/q)

http://research.att.com/~njas/sequences/A166711 for log(2)

http://research.att.com/~njas/sequences/A166871 for log(3/2) —Preceding unsigned comment added by 62.43.57.10 (talk) 09:41, 1 November 2009 (UTC)

I recently proved a special case of this (for $\log(N/1)$) and have been looking to see if it was generally known and was excited to see these links, however the seem to be broken. Does anyone have a source for this? I've also added a in the intro where this was discussed. Shawsa7 (talk) 21:49, 2 May 2019 (UTC)

These seem seem related.

http://oeis.org/A166711

http://oeis.org/A166871

Shawsa7 (talk) 22:08, 2 May 2019 (UTC)

The way that this is expressed in the intro is misleading. The way it's written now is $\ln 2=1+{\frac {1}{2}}-1+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{6}}-{\frac {1}{3}}+\dots$. This suggests the generalization - for example $\ln 4=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-1+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}-{\frac {1}{2}}+\dots$ which does not seem to be the procedure that is alluded to in the intro. This series seems to have ${\frac {1}{2}}$ and $-{\frac {1}{2}}$ whereas the method in the article only has one of them. I suggest we either remove this form and use a different one consistent with the method explained or add this method to the article with citations (looking for such citations is actually how I found myself here).Shawsa7 (talk) 00:36, 3 May 2019 (UTC)

References

I added references that discuss the Riemann-series theorem and its applications. Some of these also discuss infinite-dimensional versions (e.g. problem 28 of the Scottish Book) and links to probability theory (e.g. exchangeability, weak exchangeability, etc.). Thanks, Kiefer.Wolfowitz (talk) 17:53, 11 October 2010 (UTC)

Isn't this theorem called "rienmann-dini"?

Dini, an italian mathematician. 62.18.122.161 (talk) 14:35, 1 December 2021 (UTC)

Only in italian wikipedia. https://it.wikipedia.org/wiki/Teorema_di_Riemann-Dini 2A00:1370:8184:2478:6C69:76C3:10A3:2E5B (talk) 09:58, 5 July 2022 (UTC)

Conditional divergence?

Does not this theorem mean conditionally divergent series can be rearranged to converge, but not absolutely divergent? Obviously the defintion of absolute is different. Valery Zapolodov (talk) 04:34, 14 January 2022 (UTC)

Ha, found it! https://math.stackexchange.com/a/1221227/756502 Valery Zapolodov (talk) 04:38, 14 January 2022 (UTC)

Yes of course. But this is unimportant and certainly should not be mentioned as part of the statement of the theorem (which it is not). And it shouldn't be in the article at all without a reliable source (for reasons on due weight). --JBL (talk) 13:11, 14 January 2022 (UTC)

found Sierpinski's original papers

They are only ever published in Polish. It was hard.

The first paper was catalogued in French, but it was only published in Polish, in Sprawozdania z posiedzen Towarzystwa Naukowego Warszawskiego [Meeting reports of the Warsaw Scientific Society]. It is recorded here in page 129 of International Catalogue of Scientific Literature, 1901-1914 (1912 May, volume A Mathematics).

The title is "Przyczynek do teoryi szeregów rozbieznych", though sometimes the "teoryi" is misspelled as "teorii". I found an online version here . It was "Komunikat zgłoszony dn. 30 grudnia 1909 r." ["Communicated on 1909-12-30"].

The second paper is "Uwaga do twierdzenia Riemanna o szeregach warunkowo zbieżnych" pulished in Prace Matematyczno-Fizyczne (in Polish). 21 (1): 17–20. It comes with a French title [Remarque sur le théorème de Riemann relatif aux séries semiconvergentes], though it wasn't published in French. pony in a strange land (talk) 17:16, 31 December 2022 (UTC)