Representation of real numbers by generalized geometric series

Eugene A. Maier

Vol. 28 (1969), No. 3, 603–609

DOI: 10.2140/pjm.1969.28.603

Abstract

We shall say that the series of real numbers, ∑ _i=0^∞1∕a_i, is a generalized geometric series (g.g.s.) if and only if a_i² ≦ a_i+1a_i−1 for all i ≧ 1. (Note that the series is geometric if and only if equality holds.) In this paper we investigate the representation of positive real numbers less than or equal to one by generalized geometric series of the form ∑ _i=0^∞xⁱ∕c_i where the c_i are positive integers and x ≧ 1.

Mathematical Subject Classification

Primary: 40.10

Secondary: 10.00

Milestones

Received: 20 March 1968

Published: 1 March 1969