I recall that Riemann's theorem on conditionnaly convergent series states that if a series converges conditionnaly, then you can rearrange the sequence so that the new series converges to whatever real number you choose, or diverges.
I'm interested in Sierpinski's generalization :
https://en.wikipedia.org/wiki/Riemann_series_theorem#Sierpi.C5.84ski_theorem
Several results are stated in this paragraph and I'd like to be sure that I understand them properly. The Wikipedia article gives references but I couldn't find them and I think they're in Polish, which I don't understand.
all the indexes of the terms of the series may be rearranged
Does this mean that we can restrict ourselves to permutations with no fixed point?
Sierpiński proved that is sufficient to rearrange only some strictly positive terms or only some strictly negative terms.
So we can always choose a permutation such that every positive (resp. negative) term keeps its rank, except finitely many, and we can do what we want with negative (resp. positive) terms?
it is proved that is sufficient to rearrange only the indexes in the ideal of sets of asymptotic density zero
I don't understand this. I know what an ideal is in the context of ring theory, but what is an ideal of sets of asymptotic density zero?