Infinite series and the derived set of the aggregate of the fractional parts of its partial sums

Abstract

The aim of this paper is to study the relationship between the nature of an infinite series of real terms in which the general term tends to zero and the derived set (set of limit points) of the aggregate of the fractional parts of its partial sums. For all types of series (in which the nth term tends to zero) we determine the derived set.We denote by F the set of the fractional parts of the partial sums of the series, and by F′ the derived set of F. The principal results of the paper can be stated as follows:1. the series is convergent if and only if F has at most one limit point in [0, 1] or F′ ⊆ {0, 1};2. the series, if non-oscillatory, is divergent if and only if F′ = [0, 1];3. if the series is oscillatory, F′ is a closed sub-interval of [0, 1] or [0, 1] - (a, b), where (a, b) ⊊ (0, 1).