Involve: A Journal of Mathematics
- Volume 12, Number 2 (2019), 281-300.
Infinite sums in totally ordered abelian groups
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.
Involve, Volume 12, Number 2 (2019), 281-300.
Received: 4 August 2017
Revised: 19 January 2018
Accepted: 22 January 2018
First available in Project Euclid: 25 October 2018
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Oman, Greg; Randall, Caitlin; Robinson, Logan. Infinite sums in totally ordered abelian groups. Involve 12 (2019), no. 2, 281--300. doi:10.2140/involve.2019.12.281. https://projecteuclid.org/euclid.involve/1540432918