The mind-bending properties of infinity

1 + 2 = 2 + 1: the commutativity of addition for real numbers—the fact that the order in which terms are added has no effect on their sum—seems self-evident. Everyday experience bears this out. Yet there are a few exceptions, opening the door to some magical possibilities!

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The power to choose the sum ------------------------------

Although every sum of finitely many real numbers is indeed commutative—as are all the sums encountered in everyday life—there are situations in which the sum of infinitely many terms varies with their order. This subject, studied since the mid-19^th century, was pioneered by German mathematicians: Dirichlet seems to have been the first to establish that changing the order of the terms in certain series alters their sum; he was followed by Georg Simon Ohm (1789–1854), Bernhard Riemann and Oscar Xavier Schlömilch (1823–1901).

Bernhard Riemann discovered precise criteria that led to the rearrangement theorem. Although he was a prolific and influential pioneer of mathematics—celebrated in analysis for the integrals that now bear his name and for his research on the zeta function—Riemann received relatively little recognition for discovering this result.

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An extraordinary insight ----------------------------

Riemann proved beyond doubt that changing the order of the terms in certain sums—conditionally convergent series—can… alter their sum! His research went further still: were he here, Riemann would proclaim like a magician, "Give me any real number, or even positive or negative infinity, and I will make this series converge to that real number or make it diverge." In other words, Riemann was the first to understand how to tame such series, devising a suitable permutation of their terms to give the series any sum he chose. It is a breathtaking result!

To understand Riemann's method, let's examine his rearrangement theorem. He first describes the sequences $(u_n{)}_{n\ge 0}(u\_n)\_\{n\backslash ge\; 0\}$ to which the result applies: those satisfying the criterion for conditional convergence (that is, they converge to a real number r but diverge in absolute value). In other words, the series with general term u_n converges, while the series with general term |u_n| diverges. Formally: ${\sum }_{k=0}^n{u}_{k}r\text{and}{\sum }_{k=0}^{+\mathrm{\infty }}\mid u_k\mid =+\mathrm{\infty }\mathrm{.}\backslash sum\_\{k=0\}^\{n\}u\_k\; \backslash underset\{n\backslash to+\backslash infty\}\{\backslash longrightarrow\}\; r\; \backslash ,\backslash ,\backslash ,\backslash text\{and\}\backslash ,\backslash ,\backslash ,\; \backslash sum\_\{k=0\}^\{+\backslash infty\}\backslash left\; |\; u\_k\; \backslash right\; |\; =+\backslash infty.$

Now consider the alternating harmonic series:

This series converges to the negative of the natural logarithm of 2, namely –0.693…, whereas the harmonic series

diverges.

The series of interest to us will therefore have both strictly positive and strictly negative terms.

The conditional convergence needed to prove the rearrangement theorem is linked to the availability of infinitely many negative and positive terms. A conditionally convergent series $S={\sum }_{n=0}^{+\mathrm{\infty }}{u}_{n}S=\backslash sum\_\{n=0\}^\{+\backslash infty\}u\_n$ has infinitely many terms of each sign. For Riemann's insight to hold, the two series obtained from S by considering only its positive terms and only its negative terms must both diverge. And indeed, these two subseries do diverge. To see this, let A = (a_n)_n≥0 be the subsequence of (u_n)_n≥0 consisting solely of the positive terms, and let B = (b_n)_n≥0 be the subsequence of negative terms. We can write the terms of sequences A and B as follows:

Indeed, a_n = 0 if u_n < 0, and a_n = u_n if u_n > 0. It follows that the series whose terms are a_n is the sum of a divergent series with positive terms and a convergent series, so it diverges. Likewise, for every integer n, $b_n=\frac{u_n-\mid u_n\mid }{2}b\_n=\backslash frac\{u\_n-\backslash left\; |\; u\_n\; \backslash right\; |\}\{2\}$, and B also diverges, as the difference of a convergent series and a divergent one.

Thus, every conditionally convergent series has infinitely many positive numbers and infinitely many negative numbers, and the sums of the positive and negative terms both diverge.

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Rearranging the terms ---------------------

Take the alternating harmonic series and choose an arbitrary real number, say 10¹⁰⁰. We want to rearrange the terms of the series $-1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\dots -1+\backslash frac\{1\}\{2\}-\backslash frac\{1\}\{3\}+\backslash frac\{1\}\{4\}-\backslash frac\{1\}\{5\}+\backslash dots$ by a permutation so that our series converges not to –ln(2), but to 10¹⁰⁰.

Here is Riemann's brilliant idea: begin by adding the positive terms of the series until the sum exceeds 10¹⁰⁰. Stop as soon as 10¹⁰⁰ has been crossed; this is possible because the associated series A diverges. Now bring in B, the series of negative terms, and run through it until the sum falls below 10¹⁰⁰ again (in practice, at this stage we need only take the first term of the harmonic series, namely –1). Resume A where we left off and once again add its terms in order until the sum exceeds 10¹⁰⁰. Once 10¹⁰⁰ has been exceeded, add the second negative term of B, namely –1/3: the sum immediately falls below 10¹⁰⁰. We then resume A where we left off and continue our algorithm, and so on. No term of the harmonic series is omitted, and each term is used only once: we have therefore obtained a genuine permutation of the terms of the series under consideration. The convergence of the new series produced by rearranging the terms can be pictured as follows: a climb toward 10¹⁰⁰, followed by oscillations around 10¹⁰⁰ that grow "tighter and tighter," alternately approaching from above and below. The graph below shows the kind of sequence we construct, using the alternating harmonic series rearranged to converge to 3 as an example.

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Convergence of the rearranged alternating harmonic series to 3.

This fascinating thought experiment takes us into the mysterious world of mathematical infinity. There we encounter countable sets (the natural numbers, the integers, the rational numbers, the even numbers, the perfect squares…) and uncountable sets (the real numbers, the complex numbers, the functions from [0, 1] to itself…). The German mathematician Georg Cantor (1845–1918) founded this theory, which continues to produce extraordinarily counterintuitive results. His ideas were poorly received in his day… Today, however, more than ever, in David Hilbert's words, "No one shall expel us from the paradise that Cantor has created for us."