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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