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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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Volume 24, Issue 4


Leibniz Series for π

Karol Pąk
  • Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-02-23 | DOI: https://doi.org/10.1515/forma-2016-0023


In this article we prove the Leibniz series for π which states that π4=n=0(1)n2n+1.

The formalization follows K. Knopp [8], [1] and [6]. Leibniz’s Series for Pi is item #26 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Keywords: π approximation; Leibniz theorem; Leibniz series

MSC 2010: 40G99; 03B35


  • [1] George E. Andrews, Richard Askey, and Ranjan Roy. Special Functions. Cambridge University Press, 1999. Google Scholar

  • [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Google Scholar

  • [3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.Google Scholar

  • [4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Google Scholar

  • [5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Google Scholar

  • [6] Lokenath Debnath. The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010. Google Scholar

  • [7] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from ℝ to ℝ and integrability for continuous functions. Formalized Mathematics, 9(2):281–284, 2001.Google Scholar

  • [8] Konrad Knopp. Infinite Sequences and Series. Dover Publications, 1956. ISBN 978-0-486-60153-3. Google Scholar

  • [9] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.Google Scholar

  • [10] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471–475, 1990.Google Scholar

  • [11] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.Google Scholar

  • [12] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.Google Scholar

  • [13] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Google Scholar

  • [14] Xiquan Liang and Bing Xie. Inverse trigonometric functions arctan and arccot. Formalized Mathematics, 16(2):147–158, 2008. doi:10.2478/v10037-008-0021-3.CrossrefGoogle Scholar

  • [15] Akira Nishino and Yasunari Shidama. The Maclaurin expansions. Formalized Mathematics, 13(3):421–425, 2005.Google Scholar

  • [16] Chanapat Pacharapokin, Kanchun, and Hiroshi Yamazaki. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(2):139–141, 2004.Google Scholar

  • [17] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125–130, 1991.Google Scholar

  • [18] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.Google Scholar

  • [19] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Google Scholar

  • [20] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.Google Scholar

About the article

Received: 2016-10-18

Published Online: 2017-02-23

Published in Print: 2016-12-01

Citation Information: Formalized Mathematics, Volume 24, Issue 4, Pages 275–280, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.1515/forma-2016-0023.

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© 2016 Karol Pąk, published by De Gruyter Open. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License. BY-SA 3.0

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