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 is an element of I, there exists a series Sigma a(n) an of real terms convergent to s such that I = sigma a(p(n)) (where sigma a(p(n)) denotes the set of limit points of the partial sums of the series Sigma a(p(n))). We determine permutations p of N for which there exists a conditionally convergent series Sigma a(n) such that Sigma a(p(n)) = +infinity. If the permutation p of N possesses the last property then we prove that for any alpha is an element of R and beta is an element of R* there exists a series Sigma a(n) convergent to a and such that sigma a(p(n)) = [beta, +infinity]. We show that for any countable family P of divergent permutations there exist conditionally convergent series Sigma a(n), and Sigma b(n) such that any series of the form Sigma a(p(n)) with p is an element of P is convergent to the sum of Sigma a(n), while sigma b(p(n)) = R* for every p is an element of P. (C) 2009 Elsevier Inc. All rights reserved.