The following question was recently closed shortly after being asked:

http://mathoverflow.net/questions/47589/

Based on the comments, at least some of the closers seemed to think that it is simply the Riemann Rearrangement Theorem. But there is an additional requirement that only the negative terms of the series are permuted. In fact, if I am not mistaken, this is a 1911 theorem of W. Sierpinski (the reference is given in a comment to the question). I am reasonably sure that the proof of Sierpinski's Theorem is significantly more intricate than that of the standard Riemann Rearrangement Theorem (to the best of my knowledge, Sierpinski proved no easy theorems!).

Because of this it seems to me that the question should be reopened so that an answer can be given referring back to Sierpinski's paper as well as providing a pointer to an internet-accessible version of the proof. (I do not have such at the moment, but if the question is reopened I would be motivated to search further.) What do you think?

I was the first to vote to close, so perhaps I should say a couple of words.

First, I did not confuse the question with Riemann's theorem; rather, I found it easy to answer, so it looked to me as if the question was "too localized" (in fact, I thought it would be fine for math.SE).

It is nice that the result goes back to Sierpinski and yes, there are a few non-trivial things that can be said. In fact, there is a fairly recent (and interesting!) paper that deals with similar matters, and explicitly references Sierpinski's theorem: "Rearrangement of conditionally convergent series on a small set" by Rafał Filipów and Piotr Szuca, Journal of Mathematical Analysis and Applications, 362 (2010) 64–71.

I do not have any kind of strong stance in the matter. If it looks like people are willing to re-open it, I'm fine with it.

OK, sounds good. I didn't even have to vote to reopen it myself.