Leibniz Series for π

Abstract

In this article we prove the Leibniz series for π which states that π4=∑n=0∞(−1)n2⋅n+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/.Pąk Karol - Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandGeorge E. Andrews, Richard Askey, and Ranjan Roy. Special Functions. Cambridge University Press, 1999.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Lokenath Debnath. The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010.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.Konrad Knopp. Infinite Sequences and Series. Dover Publications, 1956. ISBN 978-0-486-60153-3.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471–475, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.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.Akira Nishino and Yasunari Shidama. The Maclaurin expansions. Formalized Mathematics, 13(3):421–425, 2005.Chanapat Pacharapokin, Kanchun, and Hiroshi Yamazaki. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(2):139–141, 2004.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125–130, 1991.Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998