Wednesday, 4 July 2018

sequences and series - Is there any geometry behind the Basel problem?



I could find many beautiful and rigorous proofs for Euler's solution to the Basel problem here Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)



Basel problem solution




But I am curious to know whether there are proofs by using geometry.



If anyone has proofs by geometry, please do share it with us.


Answer



Funny you should ask this today. A great video video by the YouTuber 3Blue1Brown was just posted today. (Aside: I recommend all his videos.)



The proof is based on the result mentioned by "3 revs" in the MO thread mentioned by user296602 above.


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