Sets of Sums of a Series Depending on Sign Distributions

Abstract

Let (an) be a sequence of positive real numbers monotonically convergent to 0 for which ∑an diverges and let E be the set of sign distributions. We call $$S(E,{a_n}) = \{ \sum\limits_{n = 1}^\infty {{\varepsilon _n}{a_n}:({\varepsilon _n})} \in E\} $$ the set of E-sums for the sequence (an). In this paper we study topological properties of sets S(EUSA, an) and S(EBSA, an), where EUSA is the set of all uniform segmentally alternating sign distributions and EBSA is the family of all bounded segmentally alternating sign distributions.