# 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)

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/A166871 for log(3/2) —Preceding unsigned comment added by 62.43.57.10 (talk) 09:41, 1 November 2009 (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)