Euler's Basel sum

This page uses the chapter's tools to prove Euler's result $\sum_{n\ge1}1/n^2=\pi^2/6$. The p-series test with $p=2$ shows that the series converges; a double integral finds its value.

The geometric-series formula and Fubini's theorem write the series as an integral over the unit square. A linear change of variables turns the square into a diamond, sliced as in regions and order of integration. Symmetry and one-variable substitutions then finish the evaluation.

Turn the series into an integral

For $n\ge0$,

$$\int_0^1x^n\,dx=\frac1{n+1}.$$

Multiplying two copies of this identity and using Fubini on the unit square gives

$$\frac1{(n+1)^2} =\int_0^1\int_0^1(xy)^n\,dx\,dy.$$

To sum over $n$, use the geometric series: for $0\le x,y<1$,

$$\sum_{n=0}^\infty(xy)^n=\frac1{1-xy}.$$

The finite geometric sums are nonnegative and increase pointwise, so monotone convergence justifies taking their limit under the integral.

The integrand $1/(1-xy)$ is singular only at the corner $(1,1)$, but it is unbounded in every neighbourhood of that corner. The integral is therefore improper, defined as the monotone limit over bounded truncations. The corner has zero area, but the blow-up near it cannot be ignored.

Summing the identities over $n\ge0$ therefore gives

$$S:=\sum_{n=1}^\infty\frac1{n^2} =\iint_{[0,1]^2}\frac1{1-xy}\,dx\,dy.$$

For the same reason, the proof applies the change-of-variables theorem on compact subregions that stay away from $(1,1)$ and then removes the truncation by monotone convergence. The transformed integrals are understood as the corresponding improper limits.