On Riemann's Rearrangement Theorem for the Alternating Harmonic Series

Publisher： Mathematical Association of America

ISSN： 1930-0972

Source： American Mathematical Monthly, Vol.117, Iss.5, 2010-05, pp. : 442-448

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Abstract

It is a well-known result by B. Riemann that the terms of a conditionally convergent series of real numbers can be rearranged in a permutation such that the resulting series converges to any prescribed sum s: add p₁ consecutive positive terms until their sum is greater than s; then subtract q₁ consecutive negative terms until the sum drops below s, and so on. For the alternating harmonic series, with the aid of a computer program, it can be noticed that there are some fascinating patterns in the sequences p_n and q_n. For example, if s = log 2 + (1/2) log (38/5) the sequence p_n is 5, 7, 8, 7, 8, 7, 8, 8, 7, 8, 7, 8, . . . in which we notice the repetition of the pattern 8, 7, 8, 7, 8, while if s = log 2+ (1/2) log (37/5) the sequence p_n is 5, 7, 7, 7, 8, 7, 8, 7, 7, 8, 7, 8, . . . in which the pattern is 7, 7, 8, 7, 8.Where do these patterns come from? Let us observe that 38/5 = 7 + 3/5 and 37/5 = 7 + 2/5. The length of the repeating pattern is the denominator 5, the values of p_n, at least from some n on, are 7 and 8, and the number 8 appears 3 times in the pattern of the first example, and 2 times in that of the second one. These are not coincidences: we explain them in this paper.