On the set of limit points of the partial sums of series rearranged by a given divergent permutation

We give a new characterization of divergent permutations. We prove that for any divergent permutation p, any closed interval I of (the 2-point compactification of ) and any real number sI, there exists a series ∑a_n of real terms convergent to s such that I=σa_p(n) (where σa_p(n) denotes the set of limit points of the partial sums of the series ∑a_p(n)). We determine permutations p of for which there exists a conditionally convergent series ∑a_n such that ∑a_p(n)=+∞. If the permutation p of possesses the last property then we prove that for any and there exists a series ∑a_n convergent to α and such that σa_p(n)=[β,+∞]. We show that for any countable family P of divergent permutations there exist conditionally convergent series ∑a_n and ∑b_n such that any series of the form ∑a_p(n) with pP is convergent to the sum of ∑a_n, while for every pP.