An Interesting New Metric and Its Applications to Alternating Series

Abstract

This paper presents one interesting metric discovered while studying convergence of series with alternating signs. In previous papers [D1] and [D2] we presented a study of 'typical' series with alternating signs. Namely, given a sequence of real nonnegative numbers whose sum is infinity we consider all different ways the signs plus or minus could be put in front of each of these numbers. With a given metric we ask what is the 'size' of the set of those choices of $+$ or $-$ for which the resulting series with alternating signs converges. The term 'size' here means either the Baire category or porosity of this set. While metrics studied before as Frèchet, Baire or Euclidean allowed us to get interesting results, they all share one undesirable property - insensitiveness to the change of a 'tail' of a sequence. The D-metric, introduced here, does not have this undesirable property.