Abstract
We show that the following general version of the Riemann–Dirichlet theorem is true: if every rearrangement of a series with pairwise commuting terms in a Hausdorff topologized semigroup converges, then its sum range is a singleton.
Highlights
In 1827, Peter Lejeune-Dirichlet was the first to notice that it is possible to rearrange the terms of certain convergent series of real numbers so that the sum changes [1]
In 1837, Dirichlet showed that this cannot happen if the series converges absolutely: if a series formed by absolute values of a term of series of real numbers converges, the series itself converges and every rearrangement converges to the same sum
It is not clear in advance that an unconditionally convergent series of real numbers is absolutely convergent, and its sum range is a singleton. This is true thanks to the following Riemann rearrangement theorem: if a convergent series of real numbers is not absolutely convergent, some rearrangement is not convergent, and its sum range is the set of all real numbers
Summary
In 1827, Peter Lejeune-Dirichlet was the first to notice that it is possible to rearrange the terms of certain convergent series of real numbers so that the sum changes [1]. It is not clear in advance that an unconditionally convergent series of real numbers is absolutely convergent, and its sum range is a singleton. This is true thanks to the following Riemann rearrangement theorem: if a convergent series of real numbers is not absolutely convergent, some rearrangement is not convergent, and its sum range is the set of all real numbers.
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