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2019 Infinite sums in totally ordered abelian groups
Greg Oman, Caitlin Randall, Logan Robinson
Involve 12(2): 281-300 (2019). 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.


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Greg Oman. Caitlin Randall. Logan Robinson. "Infinite sums in totally ordered abelian groups." Involve 12 (2) 281 - 300, 2019.


Received: 4 August 2017; Revised: 19 January 2018; Accepted: 22 January 2018; Published: 2019
First available in Project Euclid: 25 October 2018

zbMATH: 06980502
MathSciNet: MR3864218
Digital Object Identifier: 10.2140/involve.2019.12.281

Primary: 03E10 , 06F20
Secondary: 40J99

Keywords: infinite series , totally ordered abelian group , transfinite induction , transfinite recursion , well-ordered set

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.12 • No. 2 • 2019
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