Vol. 12, No. 2, 2019

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Infinite sums in totally ordered abelian groups

Greg Oman, Caitlin Randall and Logan Robinson

Vol. 12 (2019), No. 2, 281–300
DOI: 10.2140/involve.2019.12.281

The notion of convergence is absolutely fundamental in the study of calculus. In particular, it enables one to define the sum of certain infinite sets of real numbers as the limit of a sequence of partial sums, thus obtaining so-called convergent series. Convergent series, of course, play an integral role in real analysis (and, more generally, functional analysis) and the theory of differential equations. An interesting textbook problem is to show that there is no canonical way to “sum” uncountably many positive real numbers to obtain a finite (i.e., real) value. Plenty of solutions to this problem, which make strong use of the completeness property of the real line, can be found both online and in textbooks. In this note, we show that there is a more general reason for the nonfiniteness of uncountable sums. In particular, we present a canonical definition of “convergent series”, valid in any totally ordered abelian group, which extends the usual definition encountered in elementary analysis. We prove that there are convergent real series of positive numbers indexed by an arbitrary countable well-ordered set and, moreover, that any convergent series in a totally ordered abelian group indexed by an arbitrary well-ordered set has but countably many nonzero terms.

infinite series, totally ordered abelian group, transfinite induction, transfinite recursion, well-ordered set
Mathematical Subject Classification 2010
Primary: 03E10, 06F20
Secondary: 40J99
Received: 4 August 2017
Revised: 19 January 2018
Accepted: 22 January 2018
Published: 8 October 2018

Communicated by Vadim Ponomarenko
Greg Oman
Department of Mathematics
University of Colorado
Colorado Springs, CO
United States
Caitlin Randall
Department of Mathematics
University of Colorado
Colorado Springs, CO
United States
Logan Robinson
Department of Mathematics
University of Colorado
Colorado Springs, CO
United States