In 1650, Pietro Mengoli asked how much the sum of the inverse squares is worth:

$$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n^2} = \ ?$$

The Bernoullis tried and failed. The sum grows slowly: with 10 terms it gives $1{,}5498$, with 100 terms gives $1{,}6350$, with 1000 terms gives $1{,}6439$. No one recognized the number to which she walked. See the figure: each bar is the sum up to $N$ terms.

The partial sums rise more and more slowly towards π²/6. Fiddle with the controller or test a million terms.

In 1735, Euler used a bold idea. A polynomial can be written from its roots. He assumed that the same was true for the function $\dfrac{\sin x}{x}$, which is worth $1$ in $x = 0$ and cancels out in $x = \pm\pi, \pm 2\pi, \pm 3\pi, \dots$:

$$\frac{\sin x}{x} = \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right)\cdots$$

The figure shows the idea working. The blue curve is $\dfrac{\sin x}{x}$, with the roots marked in red. The curve golden is the product with just a few parentheses: increase the number of factors and see the product stick to the curve.

The more factors, the more Euler’s product looks like sin(x)/x.

If you multiply all these parentheses together, the term in $x^2$ will be

$$-\left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots\right)x^2 = -\frac{x^2}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}.$$

On the other hand, the sine Taylor series gives

$$\frac{\sin x}{x} = \frac{1}{x}\left(x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots\right) = 1 - \frac{x^2}{6} + \frac{x^4}{120} - \cdots$$

The two expressions represent the same function, so the coefficients of $x^2$ must be the same:

$$\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{1}{6} \quad\Longrightarrow\quad \sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6} \approx 1{,}644934.$$

Nobody expected to see $\pi$, a number from the geometry of the circle, in a sum of fractions. The argument Euler’s formula was not rigorous by today’s standards: writing $\frac{\sin x}{x}$ as a product infinity was only justified a century later, by Weierstrass. But the result was right, and Euler numerically checked several decimal places. With the same method he showed that $\sum \frac{1}{n^4} = \frac{\pi^4}{90}$.

All topics by Leonhard Euler · ← The language that Euler left us · The number $e$ and compound interest →

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