Relations between convergence of series and convergence of sequences

Dieter Landers and Lothar Rogge

Vol. 64 (1976), No. 2, 465–469

DOI: 10.2140/pjm.1976.64.465

Abstract

Let A = (a_n)_n∈N be a sequence of real numbers. For ξ ∈ (0,1) define

where [x] is the greatest integer less than or equal to x. If no ambiguity can arise we write S_n(ξ) instead of S_n(ξ,A). In the theory of regularly varying sequences the problem arose of concluding from the convergence of the sequence S_n(ξ), n ∈ N, for all ξ in an appropriate set K ⊂ (0,1) of real numbers, that the sequence a_n, n ∈ N, converges to zero. In this paper we give some positive results for the case that K consists of two elements.

Mathematical Subject Classification 2000

Primary: 40A05

Milestones

Received: 11 February 1976

Published: 1 June 1976