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 .