On a Typical Series with Alternating Signs

Abstract

This paper presents a study of typical series with alternating signs. Namely, given a sequence of real nonnegative numbers whose sum is infinity we consider all possible ways of placing plus or minus signs in front of each of these numbers. Choosing a convenient metric we ask what is the `size' (in terms of Baire category and porosity) of the set of those choices of $+$ or $-$ for which the resulting series converges. The author of this paper has studied this problem in his paper [D1] for the Euclidean metric. The main goal of this paper is to extend the results from [D1] for other standard metrics, such as the Fr\`echet or Baire metrics.