A proof that Euler missed: evaluating \(\zeta(2)\) the easy way. (English) Zbl 0538.10001

The author proves Euler’s identity \(\sum^{\infty}_{n=1}n^{-2} = \pi^2/6\) by simply evaluating a certain double integral in two different ways. Contrary to the more standard proofs this approach can be presented in a course in elementary calculus.


11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
40A05 Convergence and divergence of series and sequences
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[1] R. Apéry (1979) Irrationalité de {\(\zeta\)}(2) et {\(\zeta\)}(3).Astérisque 62.11–13. Paris: Société Mathématique de France.
[2] F. Beukers (1979) A note on the irrationality of {\(\zeta\)}(2) and {\(\zeta\)}(3),Bull. Lon. Math. Soc. 11:268–272. · Zbl 0421.10023 · doi:10.1112/blms/11.3.268
[3] F. Goldscheider (1913)Arch. Math. Phys. 20:323–324.
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