Remainders of power series

[摘要] Suppose∑n=0∞anznhas radius of convergenceRandσN(z)=|∑n=N∞anzn|. Suppose|z1|<|z2|σN(z) for zϵT}. Two questions are asked: (a) canSbe cofinite? (b) canSbe infinite? This paper provides some answers to these questions. The answer to (a) is no, even ifT=z2. The answer to (b) is no, forT=z2ifliman=a≠0. Examples show (b) is possible ifT=z2and forTa neighborhood ofz2.

[学科分类] 数学（综合）