ma049 - Riemanns Rearrangement Theorem Stewart Galanor, 134...

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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. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). 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 Internet: E-mail: Article reprinted with permission from Mathematics Teacher , copyright November 1987 by the National Council of Teachers of Mathematics. All rights reserved. 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 .
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Fig. 1 In 1827, Peter Lejeune-Dirichlet discovered this surprising result while working on conditions that ensured the convergence of Fourier series. He was the first to notice that it is possible to rearrange the terms of certain series (now known as conditionally convergent series) so that the sum would change. Why is this result possible? Dirichlet was never able to give an answer. (In a paper published in 1837, he did prove that rearranging the terms of an absolutely convergent series does not alter its sum.) With the discovery that the sum of a series could be changed, Dirichlet had found the path to follow to prove the convergence of Fourier series. By 1829 he had succeeded in solving one of the preeminent problems of that time. In 1852, Bernhard Riemann began work on a paper extending Dirichlet’s results on the
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ma049 - Riemanns Rearrangement Theorem Stewart Galanor, 134...

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