Riemann’s Rearrangement Theorem
Stewart Galanor, 134 West Ninety-third Street, New York, NY 10025
Mathematics Teacher,
November 1987, Volume 80, Number 8, pp. 675–681.
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Article reprinted with permission from
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S
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
S
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
and let
x
equal 1.)
It seems that although we merely rearranged the terms of an infinite series (equations 3
and 4), its sum has changed from 2
S
to
S
!
ln
s
1
1
x
d
S
5
o
‘
n
5
1
s
2
1
d
n
2
1
n
.