Chapter 15, Problem 6P

Leonhard Euler was able to find the exact sum of the series in Problem 5. In 1736 he proved that ${\displaystyle \underset{n-1}{\overset{\infty }{\sum }}\frac{1}{n^2}}=\frac{{\pi }^2}{6}$

In this problem we ask you to prove this fact by evaluating the double integral in Problem 5. Start by making the change of variables $x=\frac{u-v}{\sqrt{2}}y=\frac{u+v}{\sqrt{2}}$

This gives a rotation about the origin through the angle $\pi /4$. You will need to sketch the corresponding region in the uv-plane.

[Hint: If, in evaluating the integral, you encounter either of the expressions (1 – sin θ)/cos θ or (cos θ)/(1 + sin θ), you might like to use the identity cos θ = sin(( $\pi /2$) − θ) and the corresponding identity for sin θ.]