A Proof of ζ(2)=π²/6 Involving a Fourier–Legendre Expansion

Abstract

We give a new proof of Euler’s famous formula $\zeta (2)={\pi }^2/6$ using a 1954 result by M. O. González which expands the complete elliptic integral of the first kind in terms of Legendre polynomials.

MSC:

• 11Y60

Acknowledgment

We are grateful for Editorial/Reviewer feedback that has been provided and that has greatly improved the exposition and structure of this note.

Notes

1 See https://mathworld.wolfram.com/MaclaurinSeries.html, for example.