Basel Problem

Pąk, Karol; Korniłowicz, Artur

doi:10.1515/forma-2017-0014

Abstract

A rigorous elementary proof of the Basel problem [6, 1] $\sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{π^{2}}{6}$ $$\sum\nolimits_{n = 1}^\infty {{1 \over {n^2 }} = {{\pi ^2 } \over 6}} $$ is formalized in the Mizar system [3]. This theorem is item #14 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

References

[1] M. Aigner and G. M. Ziegler. Proofs from THE BOOK. Springer-Verlag, Berlin Heidelberg New York, 2004.10.1007/978-3-662-05412-3
Search in Google Scholar Back to article
[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
Search in Google Scholar Back to article
[3] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-817.10.1007/978-3-319-20615-817
Open DOI Search in Google Scholar Back to article
[4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.
Search in Google Scholar Back to article
[5] Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.
Search in Google Scholar Back to article
[6] Augustin Louis Cauchy. Cours d’analyse de l’Ecole royale polytechnique. de l’Imprimerie royale, 1821.
Search in Google Scholar Back to article
[7] Artur Korniłowicz and Karol Pąk. Basel problem – preliminaries. Formalized Mathematics, 25(2):141–147, 2017. doi: 10.1515/forma-2017-0013.10.1515/forma-2017-0013
Open DOI Search in Google Scholar Back to article
[8] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265–269, 2001.
Search in Google Scholar Back to article
[9] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.
Search in Google Scholar Back to article
[10] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391–395, 2001.
Search in Google Scholar Back to article
[11] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461–470, 2001.
Search in Google Scholar Back to article
[12] Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1): 49–58, 2004.
Search in Google Scholar Back to article
[13] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.
Search in Google Scholar Back to article
[14] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.
Search in Google Scholar Back to article
[15] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.
Search in Google Scholar Back to article

Basel Problem

Abstract

Paradigm

My account