Are there any visual proofs for $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$?

Question

I was flipping through Proofs Without Words (PWW) and saw many visual proofs for sequences and series. However, I saw none for $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$

Are there any visual proofs for the above series?

Answer 1

If you know what Fourier Series are, I think you may want to start this problem by looking at it as a Fourier series, which will allow a more mathematically rigorous proof to show the Basel Problem which was solved by Euler. If not there is another, more elementary way to do it as well at this link:

http://en.wikipedia.org/wiki/Basel_problem

It goes through the proof numerous ways, but the Fourier series and another way are the most mathematically rigorous proofs (Sections 3 & 4).

Answer 2

In his extraordinary paper, Mikael Passare presents following visual idea:

Even more amazing than the above picture are techniques used for the proof. They involve basic math only, essentially trigonometry and more visual transformations of curved (sometimes infinite!) and straight line areas, like this one:

Here all six region have the same area, check the details in the paper.