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

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?

• You can try this answer or this paper. Nov 29, 2014 at 19:28
• @Nameless, great finds Nov 29, 2014 at 19:40
• Yes, there are a visual proofs for $\sum_{k=1}^{+\infty }\,\frac{1}{n^2}=\frac{\pi ^2}{6}$. Please check this pdf from Les-Mathematiques.net (registration required). Also you can check Masayoshi Hata's Problems And Solutions In Real Analysis which gives 13 elementary proofs for $\sum_{k=1}^{+\infty }\,\frac{1}{n^2}=\frac{\pi ^2}{6}$. Nov 29, 2014 at 20:07
• This is related to the Basel problem: en.wikipedia.org/wiki/… Here is something like what you are asking for: demonstrations.wolfram.com/PackingSquaresWithSide1N Feb 21, 2015 at 22:51
• @Nameless Why don't you post that an answer? Or shall I do it with your request? It seems to answer the question, and the comments say avoid answering... Mar 1, 2015 at 4:51

## 3 Answers

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.

Grant Sanderson just published a video on his channel 3Blue1Brown in which he presents a very accessible way of visualizing the problem and its solution. Link to the video:

https://youtu.be/d-o3eB9sfls

• Please try to describe as much here as possible in order to make the answer self-contained. Links are fine as support, but they can go stale and then an answer which is nothing more than a link loses its value. Please read this post.
– robjohn
Mar 3, 2018 at 20:49
• Adding a link doesn't actually answer any question... :-)
– hola
Dec 1, 2019 at 14:38
• @xxx--- feel free to flag my answer if you feel it should not belong here :-) Dec 1, 2019 at 19:37
• Just relax. :-)
– hola
Dec 2, 2019 at 7:17

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).