The infinite decimal expansion of a number like is another example of an infinite set of numbers whose sum is finite. The nth decimal place tells me how many 1/10ⁿ-ths to add to get closer to an expression for . Euler had spent some time fascinated by adding together infinite lists of numbers to see if they might actually home in on some number, or whether they spiral off to infinity. One of the things which made Euler famous was a rather beautiful but unexpected infinite sum that he discovered involving which had none of the randomness of the decimal expansion.

The origin of his formula was an infinite sum that people had known for some time did not add up to a nice number but just got bigger the more terms you included, namely:

1+1/2+1/3+1/4+1/5+…

This is called the harmonic series because each term represents the wavelength of successive harmonics of a vibrating string. Looking at how this sum grows as one adds more fractions, Euler might have been mistaken for thinking it never got above 100:

1+1/2+1/3+1/4+…+1/100=5.18738…
1+1/2+1/3+1/4+…+1/1000=7.48547…
1+1/2+1/3+1/4+…+1/10000=9.78760…

But as a student Euler had learnt about a recipe first concocted in the mid fourteenth century to show why this sum must eventually spiral off to infinity. It told Euler to begin by taking the first 9 fractions: 1,1/2,…,1/9. Each fraction is bigger than 1/10 so their sum is more than 9/10. He then needed to take the next 90 fractions: 1/10,1/11,…,1/99. This time each fraction is bigger than 1/100. So these 90 numbers add up to something more than 90/100 which is just 9/10 again. But the recipe told Euler that he could just keep on doing this taking larger batches which gave him a sum greater than 9/10 each time. So he could get the sum as big as he wanted by adding up enough terms in the expression. It must be said that you have to take a heck of a lot of terms to get anywhere. To get the sum above 200 requires taking 10⁸⁶ fractions, more than there are atoms in the universe. It is only by theoretical analysis that ultimately Euler knew that this sum would pass every number. Again the power of proof over relying on experimental evidence.

In fact the speed which this sum creeps up is showing quite a strong similarity to the logarithm function again. Each time Euler wanted to add another 9/10, he needed to increase the number of fractions to achieve this by a factor of ten each time: 10 fractions for the first 9/10, 100 fractions for the next 9/10, 1000 fractions for the next. Again this additive - multiplicative connection is indicative of some connection with logarithms. So this is what Euler would have meant by writing

1+1/2+1/3+1/4+1/5+…=log

expressing the fact that the way this sum tends to infinity is like the way the number log(N) tends to infinity. So adding N terms in the expression has got Euler about as far as log(N) on the way to infinity.

Euler had been picked up as a student by the famous Bernoulli family which boasted three generations of mathematicians. Their attention had saved Euler from his father's wish for him to succeed him as pastor of their small home town in Switzerland. One of the problems that the Bernoullis amongst others were intrigued by was whether they could identify the infinite sum where one took the square of each term in the harmonic series:

1²+1/2²+1/3²+1/4²…=1+1/4+1/9+1/16+…

This was by Euler's time a classical problem that had stumped the best calculators of the age. It is certainly smaller than the sum of the harmonic series 1+1/2+1/3+… since we are now adding together a selection of fractions from the original sum. It is possible that this sum would also spiral off to infinity but a little thought reveals that it can't get further than the infinite sum in Zeno's paradox.

Why is Euler's sum less than Zeno's?

Euler's sum: 1+1/2²+1/3²+1/4²+1/5²+1/6²+1/7²+…=1+1/4+1/9+1/16+1/25+1/36+1/49+…
Zeno's sum:
1+1/2+1/4+1/8+1/16+1/32+1/64+…

Your first guess might be that the opposite is true. The first six fractions in Euler's sum admittedly are each less than the corresponding fractions in Zeno's sum. But if you compare the 7th fraction, Euler's has gone in the lead. In fact every fractions after the 7th, Euler's leads Zeno's. But let's show that the lead that Zeno got over the first six terms can't actually get overturned.

The first number in both sums is 1. Take the next two fractions of Euler's sum 1/2²+1/3². This adds up to less than 1/2²+1/2²=1/2, the second fraction in Zeno's sum. The next four fractions in Euler's sum 1/4²+1/5²+1/6²+1/7² add up to less than 1/4²+1/4²+1/4²+1/4² which is 4/4²=1/4, the next fraction in Zeno's sum. Similarly the next 8 fractions in Euler's sum add up to less than 1/8. So Euler's sum can't climb higher than Zeno's.

Daniel Bernoulli had estimated the sum of the squares of the harmonic series to be "very nearly 8/5" in a letter to Goldbach dated 1728. Euler had recently joined Daniel Bernoulli in the Academy in St. Petersburg. It was not long before the series began to intrigue the young Euler. As he wrote in 1735 "So much work has been done on the series that it seems hardly likely that anything new about them may still turn up…I, too, in spite of repeated effort, could achieve nothing more than approximate values for their sums…Now, however, quite unexpectedly, I have found an elegant formula depending upon the quadrature of the circle", in modern parlance on the number .

By some pretty reckless analysis Euler had discovered that this infinite sum was homing in on the number ²/6:

1+1/4+1/9+1/16+…=²/6.

To this day, this ranks as one of the most intriguing calculations and it certainly took the scientific community of Euler's time by storm. No one would have predicted a link between this innocent sum and the number . Whereas the decimal expansion of ² is completely chaotic, Euler had discovered an alternative expansion which has as strong a pattern as one could imagine. The movie hints at some cabalistic connections that are revealed in the decimal expansion of . In my view, Euler's discovery is much spookier than connections with Jewish mysticism.

Within a few years Euler had managed to identify the sum of every even power of the harmonic series, namely if n is an even number then Euler showed that

1+1/2ⁿ+1/3ⁿ +…

was equal to some fraction times the nth power of . And in homage to the Bernoulli family whose support had brought him to St. Petersburg, he even managed to identify the fractions as special numbers that the Bernoulli family had championed known to this day as the Bernoulli numbers. Discovered by Daniel's father Jacob, these numbers have an uncanny way of cropping up in a whole host of problems. Indeed after Kummer realised in the mid eighteenth century that he had not solved Fermat's Last Theorem, the equations xⁿ+yⁿ=zⁿ for which his ideas did work depended on these Bernoulli numbers. Although Euler had identified the sum of even powers of the harmonic series, an expression for the sum if n is odd remains a mystery even today.

Euler had succeeded in finding values for:

1+1/2²+1/3²+…
1+1/2⁴+1/3⁴+…

and every even power of the harmonic series. To this day, the odd powers remain uncalculated although we do know that they must home in on some number, we just don't know which one.

Even in Euler's time, mathematicians were aware that there was an interesting function hiding behind these infinite sums. If you take any number x, the function calculates the infinite sum:

1+1/2^x+1/3^x+…

Euler had been interested in the answer when x was a whole number like 2, 3 or 4. But mathematicians realised that whenever x was bigger than 1, this sum should home in on some fixed number. This number mathematicians denoted by \zeta(x) and the function became known as the zeta function. So Euler's famous calculation can be written as z (2)=²/6.

The graph of the zeta function for numbers x bigger than 1 is a nice smooth function. But looking at it, you wouldn't think it had much potential to tell you anything interesting.