International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 2, Pages 239-250
doi:10.1155/S0161171279000223

Remainders of power series

J. D. McCall,1 G. H. Fricke,2 and W. A. Beyer3

1Department of Mathematics, LeMoyne–Owen College, Memphis 38126, Tennessee, USA
2Department of Mathematics, Wright State University, Dayton 45431, Ohio, USA
3Department of Mathematics, Los Alamos Scientific Laboratory, Los Alamos 87545, New Mexico, USA

Received 21 February 1978; Revised 20 March 1979

Copyright © 1979 J. D. McCall et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Suppose ∑n=0∞anzn has radius of convergence R and σN(z)=|∑n=N∞anzn|. Suppose |z1|<|z2|<R, and T is either z2 or a neighborhood of z2. Put S={N|σN(z1)>σN(z) for zϵT}. Two questions are asked: (a) can S be cofinite? (b) can S be infinite? This paper provides some answers to these questions. The answer to (a) is no, even if T=z2. The answer to (b) is no, for T=z2 if liman=a≠0. Examples show (b) is possible if T=z2 and for T a neighborhood of z2.