Rearranging the Alternating Harmonic Series

By Larry Riddle, Agnes Scott College

Why are conditionally convergent series interesting? While mathematicians might undoubtably give many answers to such a question, Riemann's theorem on rearrangements of conditionally convergent series would probably rank near the top of most responses. Conditionally convergent series are those series that converge as written, but do not converge when each of their terms is replaced by the corresponding absolute value. The nineteenth-century mathematician Georg Friedrich Bernhard Riemann (1826-1866) proved that such series could be rearranged to converge to any prescribed sum. Almost every calculus text contains a chapter on infinite series that distinguishes between absolutely and conditionally convergent series. Students see the usefulness of studying absolutely convergent series since most convergence tests are for positive series, but to them conditionally convergent series seem to exist simply to provide good test questions for the instructor. This is unfortunate since the proof of Riemann's theorem is a model of clever simplicity that produces an exact algorithm. It is clear, however, that even with such a simple example as the alternating harmonic series one cannot hope for a closed form solution to the problem of rearranging it to sum to an arbitrary real number. Nevertheless, it is an old result that for any real number of the form ln(sqrt(r)), where r is a rational number, there is a very simple rearrangement of the alternating harmonic series having this sum. Moreover, this rearrangement shares, at least in the long run, a nice feature of Riemann's rearrangement, namely that the partial sums at the breaks between blocks of different sign oscillate around the sum of the series.

If we rearrange the alternating harmonic series so that instead of alternating 1 positive odd reciprocal with 1 even reciprocal, we alternate blocks of 375 consecutive odd reciprocals (the positive terms) with blocks of 203 consecutive even reciprocals (the negative terms), we will get a series that converges to 1 (well, almost) as depicted in Figure 1 (the vertical axis is not to scale in this figure). The turns in Figure 1 indicate the partial sums at the break between the positive and negative blocks. The partial sums seem to oscillate around 1. Now if we rearrange the series with blocks of 1 positive term followed by blocks of 36 negative terms, we will get a series that converges exactly to -ln(3). However, as we see from Figure 2, for this rearrangement the partial sums at the breaks require some initial posturing before they appear to start oscillating around the sum.

figure 1
Figure 1: p = 375, n = 203 and Figure 2: p = 1, n = 36

How did we know to choose 375 and 203 in the first example, or 1 and 36 in the second? Why does the first example almost converge to 1, but the second converges exactly to -ln(3)? Why do these partial sums converge in the manner depicted in Figures 2 and 3? The purpose of this article is to address these issues by considering the following questions:

The latter two questions are completely answered by Riemann's theorem for rearrangements of arbitrary conditionally convergent series, but our goal is to provide a more concrete setting of Riemann's results within the context of the alternating harmonic series with the hope that the reader will then have a better understanding of the general theory. As a bonus, we will encounter along the way some interesting mathematics in the form of Riemann sums, Euler's constant, continued fractions, geometric sums, graphing via derivatives, and even Descarte's Rule of Signs.

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Last Modified: August 2000