Riemann’s Rearrangement Theorem
Stewart Galanor, 134 West Ninety-third Street, New York, NY 10025
November 1987, Volume 80, Number 8, pp. 675–681.
is a publication of the National Council of Teachers of
More than 200 books, videos, software, posters, and research reports are available
through NCTM’s publication program. Individual members receive a 20% reduction off
the list price.
For more information on membership in the NCTM, please call or write:
NCTM Headquarters Office
1906 Association Drive
Reston, Virginia 20191-9988
Phone: (703) 620-9840
Fax: (703) 476-2970
Article reprinted with permission from
, copyright November 1987
by the National Council of Teachers of Mathematics. All rights reserved.
ince that ancient time when Zeno first sent Achilles chasing after the tortoise,
infinite series have been a source of wonder and amusement because they can be
manipulated to appear to contradict our understanding of numbers and nature.
Zeno’s paradoxes still intrigue and baffle us even though the fallacies in his arguments
have long since been identified.
Mathematicians of the late seventeenth and eighteenth centuries were often puzzled by
the results they would get while working with infinite series. By the nineteenth century
it had become apparent that divergent series were often the cause of the difficulties.
“Divergent series are the invention of the devil,” Neils Hendrik Abel wrote in a letter to
a friend in 1826. “By using them, one may draw any conclusion he pleases, and that is
why these series have produced so many fallacies and so many paradoxes” (Kline 1972).
As an example of what can go wrong, suppose we let
represent the sum of the
alternating harmonic series, that is
See figure 1. What’s wrong here? (This series, by the way, is not divergent. Its sum is
ln 2, which is easy to determine. Find the Taylor series of
It seems that although we merely rearranged the terms of an infinite series (equations 3
and 4), its sum has changed from 2