  Reordering the terms of a series is a useful mathematical device, and much is
known about when it can be done without affecting the convergence or the sum of
the series. For example, if a series of real numbers absolutely converges, we
can add the even-indexed and odd-indexed terms separately, or arrange the terms
in an infinite two-dimensional table and first compute the sum of each column.
The possibility of even more intricate re-orderings prompts us to find a
general underlying principle. We identify such a principle in the setting of
Banach spaces, where we consider well-ordered series with indices beyond
{\omega}, but strictly under {\omega}_1 . We prove that for every absolutely
convergent well-ordered series indexed by a countable ordinal, if the series is
rearranged according to any countable ordinal, then the absolute convergence
and the sum of the series remain unchanged.
