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 R* (the 2-point compactification of R) and any real number s∈I, there exists a series ∑an of real terms convergent to s such that I=σap(n) (where σap(n) denotes the set of limit points of the partial sums of the series ∑ap(n)). We determine permutations p of N for which there exists a conditionally convergent series ∑an such that ∑ap(n)=+∞. If the permutation p of N possesses the last property then we prove that for any α∈R and β∈R* there exists a series ∑an convergent to α and such that σap(n)=[β,+∞]. We show that for any countable family P of divergent permutations there exist conditionally convergent series ∑an and ∑bn such that any series of the form ∑ap(n) with p∈P is convergent to the sum of ∑an, while σbp(n)=R* for every p∈P.