Vol. 64, No. 2, 1976

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Relations between convergence of series and convergence of sequences

Dieter Landers and Lothar Rogge

Vol. 64 (1976), No. 2, 465–469

Let A = (an)nN be a sequence of real numbers. For ξ (0,1) define

Sn (ξ,A) :=        ak,  n ∈ N

where [x] is the greatest integer less than or equal to x. If no ambiguity can arise we write Sn(ξ) instead of Sn(ξ,A). In the theory of regularly varying sequences the problem arose of concluding from the convergence of the sequence Sn(ξ), n N, for all ξ in an appropriate set K (0,1) of real numbers, that the sequence an, 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
Received: 11 February 1976
Published: 1 June 1976
Dieter Landers
Lothar Rogge