The Wednesday Morning Math Challenge: Week 8 Answers

In Week 8, we only tackled a problem so big — and, to me, so interesting — that it deserved its own week. For those who haven’t gotten a chance to try it yet, just click here.

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The sum

1 – 1/2 – 1/4 + 1/3 – 1/6 – 1/8 + 1/5 – 1/10 – 1/12 + 1/7 – 1/14 – 1/16 + 1/9 – 1/18 – 1/20 + …..

is equal to half the sum

1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + 1/9 – 1/10 + ….

(or half the natural log of two).

This result might strike you as crazy. It shocked me the first time I saw it. After all, one of the fundamental rules of arithmetic — known as the commutative law — means that you can switch around the numbers you’re adding as much as you want and still get the same result. (For instance, 1 + (-2) + 3 = 2, and 3 + 1 + (-2) = 2.) But here, we’re dealing with infinite sums, and infinite sums sometimes violate certain properties. The rules don’t always apply in quite the way we’re used to.

Let’s approach the sum by looking at in chunks. The sum of the first two terms (1 and -1/2) equals 1/2, so the first three terms are equivalent to

1/2 – 1/4.

The sum of the fourth term (1/3) and the fifth (-1/6) equals 1/6, so we can write the beginning of the series as

1/2 – 1/4 + 1/6 – 1/8.

The sum of the seventh term (1/5) and eighth (-1/10) equals 1/10, so now we have

1/2 – 1/4 + 1/6 – 1/8 + 1/10.

You can keep going like this — forever! We will get the sum

1/2 – 1/4 + 1/6 – 1/8 + 1/10 – 1/12 + 1/14 – 1/16 + 1/18 – 1/20 + …

But soon, you’ll start to notice something.

Look at the original sum, which is known as the alternating harmonic series: 1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + 1/9 – 1/10 + …

We see that this new sum

1/2 – 1/4 + 1/6 – 1/8 + 1/10 – 1/12 + 1/14 – 1/16 + 1/18 – 1/20 + …

is simply half of the alternating series!

It turns out that you can shuffle that series not only so that the sum equals 1/2 the natural log of two, but so that it equals any real number you choose. In the nineteenth century, a brilliant German mathematician named Bernhard Riemann showed that if an infinite series is conditionally convergent, it can be reordered so that the resulting series converges to any given real number. His theorem is known as the Riemann Rearrangement Theorem.

The Riemann Rearrangement Theorem encapsulates so much of what I love about math. It’s simple, but it challenges intuition. It shows the world to be both stranger and more elegant than it first seems.