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 ∑ a_n of real terms convergent to s such that I = σ a_{p (n)} (where σ a_{p (n)} denotes the set of limit points of the partial sums of the series ∑ a_{p (n)}). We determine permutations p of N for which there exists a conditionally convergent series ∑ a_n such that ∑ a_{p (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 ∑ a_n convergent to α and such that σ a_{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 p ∈ P is convergent to the sum of ∑ a_n, while σ b_{p (n)} = R^* for every p ∈ P.