One thing that's great about maths is that it can cope with the tricky concept of infinity. An example of this comes from infinite sums, also known as infinite series.

An example is $$\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\dots$$

As the dots suggest, the series goes on for ever, with each new term four times smaller than the previous one.

Yet the "sum" is finite, in fact, we can say that it's equal to $\frac{1}{3}$. I would now like to prove this in a slightly unconventional way . . .using chocolate.

Imagine, if you will, that you are a shopkeeper, and selling chocolate bars on the special offer shown below.

Now, each bar consists of 1 unit of actual chocolate and 1 coupon. And a customer with 4 coupons can claim 1 bar free. The key to proving our result turns out to be the following question:

What is one coupon worth, in terms of actual chocolate?

In search of an infinite series

Well, 4 coupons are worth 1 bar, and therefore 1 coupon is, in a sense, worth $\frac{1}{4}$ bar. So 1 coupon is worth $$\frac{1}{4}[\text{chocolate}+\text{coupon}],$$ where "chocolate" here is just my shorthand for the amount of actual chocolate in one bar. Now, the coupon in brackets is itself worth $\frac{1}{4}$ bar, so 1 coupon can also be said to be worth $$\frac{1}{4}[\text{chocolate}+\frac{1}{4}(\text{chocolate}+\text{coupon})].$$ And if we continue like this for ever, we find that 1 coupon is worth $$(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\dots )$$ units of actual chocolate. To prove our result, then, we just need to prove that 1 coupon is also worth $\frac{1}{3}$ unit of actual chocolate.

In search of 1/3

To put it another way, we need to prove that 3 coupons are worth 1 unit of actual chocolate. And to do this, imagine that I have saved up 3 coupons, come to your shop, and say:

How much chocolate do you get for a coupon?

"Good morning. I should like to buy a bar of chocolate. I intend to eat the chocolate straight away, in your shop, and I will pay you when I have eaten it."

Your first thought will doubtless be "another crazy mathematician!" but, being of generous spirit, you give me a bar of chocolate.

I open it, and eat the chocolate.

Next, I take out the coupon that's inside, and add it to the 3 that I have already, so that I now have 4 coupons!

I then present these to you, saying: "Here are 4 coupons. I claim a bar of chocolate!"

At which point, you say: "You've got a short memory, haven't you? I've just given you a bar of chocolate!"

Somewhat embarrassed, I reply: "Of course you have. How silly of me. So you don't owe me anything, and I don't owe you anything, either. Our business is complete. Thank you very much. Good morning."

Now, I arrived with 3 coupons and no chocolate, and went away with no coupons and 1 unit of chocolate. So 1 coupon is worth $\frac{1}{3}$ unit of chocolate, and therefore $$\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\dots =\frac{1}{3}.$$

And whatever we may think of such an unusual proof, the result itself contains a very important idea: it is possible for infinitely many positive numbers to have a finite sum.

Further reading

To find out more about the wonderful concept of infinite series, how to cope with them, and how they can turn our tricky, see these Plus articles.

About this article

This article is an excerpt of David Acheson's new book The spirit of mathematics, published by Oxford University Press in March 2023, available in hardback and eBook formats, £14.99. © David Acheson.

David Acheson is Emeritus Fellow of Jesus College, Oxford, and was the University's first winner of a National Teaching Fellowship in 2004. He was President of the Mathematical Association from 2010 to 2011, and now lectures widely on mathematics to young people and the general public. In 2013, Acheson was awarded an Honorary D.Sc. by the University of East Anglia for his outstanding work in the popularisation of mathematics. His books include 1089 and All That (OUP, 2002), The Calculus Story, (OUP, 2017), and The Wonder Book of Geometry, (OUP, 2020).