A003881 Decimal expansion of Pi/4.
7, 8, 5, 3, 9, 8, 1, 6, 3, 3, 9, 7, 4, 4, 8, 3, 0, 9, 6, 1, 5, 6, 6, 0, 8, 4, 5, 8, 1, 9, 8, 7, 5, 7, 2, 1, 0, 4, 9, 2, 9, 2, 3, 4, 9, 8, 4, 3, 7, 7, 6, 4, 5, 5, 2, 4, 3, 7, 3, 6, 1, 4, 8, 0, 7, 6, 9, 5, 4, 1, 0, 1, 5, 7, 1, 5, 5, 2, 2, 4, 9, 6, 5, 7, 0, 0, 8, 7, 0, 6, 3, 3, 5, 5, 2, 9, 2, 6, 6, 9, 9, 5, 5, 3, 7
Offset: 0
Examples
0.785398163397448309615660845819875721049292349843776455243736148...
N = 2, m = 6: Pi/4 = 4!*3^4 Sum_{k >= 0} (-1)^k/((2*k - 11)*(2*k - 5)*(2*k + 1)*(2*k + 7)*(2*k + 13)). - _Peter Bala_, Nov 15 2016
References
- Jörg Arndt and Christoph Haenel, Pi: Algorithmen, Computer, Arithmetik, Springer 2000, p. 150.
- Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 437.
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 6.3 and 8.4, pp. 429 and 492.
- Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, p. 408.
- J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 136.
- James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 119.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Sanjar M. Abrarov and Brendan M. Quine, A Viète-like formula for pi based on infinite sum of the arctangent functions with nested radicals, figshare, 4509014, (2017).
- Peter Bala, Arctanh(z) and the Legendre polynomials
- Jonathan M. Borwein, Peter B. Borwein, and Karl Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
- Antonio Gracia Llorente, Shifting Constants Through Infinite Product Transformations, OSF Preprint, 2024.
- Ronald K. Hoeflin, Titan Test.
- Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
- Literate Programs, Pi with Machin's formula (Haskell).
- Michael Penn, A surprising appearance of pie!, YouTube video, 2020.
- Michael Penn, Transforming normal identities into "crazy" ones, YouTube video, 2022.
- Srinivasa Ramanujan, Question 353, J. Ind. Math. Soc.
- Eric Weisstein's World of Mathematics, Prime Products.
- Wikipedia, Leibniz formula for Pi.
- Index entries for sequences from "Goedel, Escher, Bach".
- Index entries for transcendental numbers.
Crossrefs
Programs
-
Haskell
-- see link: Literate Programs import Data.Char (digitToInt) a003881_list len = map digitToInt $ show $ machin `div` (10 ^ 10) where machin = 4 * arccot 5 unity - arccot 239 unity unity = 10 ^ (len + 10) arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where arccot' x unity summa xpow n sign | term == 0 = summa | otherwise = arccot' x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign) where term = xpow `div` n -- Reinhard Zumkeller, Nov 20 2012 -
Magma
R:= RealField(100); Pi(R)/4; // G. C. Greubel, Mar 08 2018
-
Maple
evalf(Pi/4) ;
-
Mathematica
RealDigits[N[Pi/4,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *) (* PROGRAM STARTS *) (* Define the nested radicals a_k by recurrence *) a[k_] := Nest[Sqrt[2 + #1] & , 0, k] (* Example of Pi/4 approximation at K = 100 *) Print["The actual value of Pi/4 is"] N[Pi/4, 40] Print["At K = 100 the approximated value of Pi/4 is"] K := 100; (* the truncating integer *) N[Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 40] (* equation (8) *) (* Error terms for Pi/4 approximations *) Print["Error terms for Pi/4"] k := 1; (* initial value of the index k *) K := 10; (* initial value of the truncating integer K *) sqn := {}; (* initiate the sequence *) AppendTo[sqn, {"Truncating integer K ", " Error term in Pi/4"}]; While[K <= 30, AppendTo[sqn, {K, N[Pi/4 - Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 1000] // N}]; K++] Print[MatrixForm[sqn]] (* Sanjar Abrarov, Jan 09 2017 *)
-
PARI
Pi/4 \\ Charles R Greathouse IV, Jul 07 2014
-
SageMath
# Leibniz/Cohen/Villegas/Zagier/Arndt/Haenel def FastLeibniz(n): b = 2^(2*n-1); c = b; s = 0 for k in range(n-1,-1,-1): t = 2*k+1 s = s + c/t if is_even(k) else s - c/t b *= (t*(k+1))/(2*(n-k)*(n+k)) c += b return s/c A003881 = RealField(3333)(FastLeibniz(1330)) print(A003881) # Peter Luschny, Nov 20 2012
Formula
Equals Integral_{x=0..oo} sin(2x)/(2x) dx.
Equals Integral_{x=0..1} 1/(1+x^2) dx. - Gary W. Adamson, Jun 22 2003
Equals Integral_{x=0..Pi/2} sin(x)^2 dx, or Integral_{x=0..Pi/2} cos(x)^2 dx. - Jean-François Alcover, Mar 26 2013
Equals (Sum_{x=0..oo} sin(x)*cos(x)/x) - 1/2. - Bruno Berselli, May 13 2013
Equals (-digamma(1/4) + digamma(3/4))/4. - Jean-François Alcover, May 31 2013
Equals Sum_{n>=0} (-1)^n/(2*n+1). - Geoffrey Critzer, Nov 03 2013
Equals Integral_{x=0..1} Product_{k>=1} (1-x^(8*k))^3 dx [cf. A258414]. - Vaclav Kotesovec, May 30 2015
Equals Product_{k in A071904} (if k mod 4 = 1 then (k-1)/(k+1)) else (if k mod 4 = 3 then (k+1)/(k-1)). - Dimitris Valianatos, Oct 05 2016
From Peter Bala, Nov 15 2016: (Start)
For N even: 2*(Pi/4 - Sum_{k = 1..N/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^(N/2)*(1/N - 1/N^3 + 5/N^5 - 61/N^7 + 1385/N^9 - ...), where the sequence of unsigned coefficients [1, 1, 5, 61, 1385, ...] is A000364. See Borwein et al., Theorem 1 (a).
For N odd: 2*(Pi/4 - Sum_{k = 1..(N-1)/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^((N-1)/2)*(1/N - 1/N^2 + 2/N^4 - 16/N^6 + 272/N^8 - ...), where the sequence of unsigned coefficients [1, 1, 2, 16, 272, ...] is A000182 with an extra initial term of 1.
For N = 0,1,2,... and m = 1,3,5,... there holds Pi/4 = (2*N)! * m^(2*N) * Sum_{k >= 0} ( (-1)^(N+k) * 1/Product_{j = -N..N} (2*k + 1 + 2*m*j) ); when N = 0 we get the Madhava-Gregory-Leibniz series for Pi/4.
For examples of asymptotic expansions for the tails of these series representations for Pi/4 see A024235 (case N = 1, m = 1), A278080 (case N = 2, m = 1) and A278195 (case N = 3, m = 1).
For N = 0,1,2,..., Pi/4 = 4^(N-1)*N!/(2*N)! * Sum_{k >= 0} 2^(k+1)*(k + N)!* (k + 2*N)!/(2*k + 2*N + 1)!, follows by applying Euler's series transformation to the above series representation for Pi/4 in the case m = 1. (End)
From Peter Bala, Nov 05 2019: (Start)
For k = 0,1,2,..., Pi/4 = k!*Sum_{n = -oo..oo} 1/((4*n+1)*(4*n+3)* ...*(4*n+2*k+1)), where Sum_{n = -oo..oo} f(n) is understood as lim_{j -> oo} Sum_{n = -j..j} f(n).
Equals Integral_{x = 0..oo} sin(x)^4/x^2 dx = Sum_{n >= 1} sin(n)^4/n^2, by the Abel-Plana formula.
Equals Integral_{x = 0..oo} sin(x)^3/x dx = Sum_{n >= 1} sin(n)^3/n, by the Abel-Plana formula. (End)
From Amiram Eldar, Aug 19 2020: (Start)
Equals arcsin(1/sqrt(2)).
Equals Product_{k>=1} (1 - 1/(2*k+1)^2).
Equals Integral_{x=0..oo} x/(x^4 + 1) dx.
Equals Integral_{x=0..oo} 1/(x^2 + 4) dx. (End)
With offset 1, equals 5 * Pi / 2. - Sean A. Irvine, Aug 19 2021
Equals (1/2)!^2 = Gamma(3/2)^2. - Gary W. Adamson, Aug 23 2021
Equals Integral_{x = 0..oo} exp(-x)*sin(x)/x dx (see Rivaud reference). - Bernard Schott, Jan 28 2022
From Amiram Eldar, Nov 06 2023: (Start)
Equals beta(1), where beta is the Dirichlet beta function.
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p)^(-1). (End)
Equals arctan( F(1)/F(4) ) + arctan( F(2)/F(3) ), where F(1), F(2), F(3), and F(4) are any four consecutive Fibonacci numbers. - Gary W. Adamson, Mar 03 2024
Pi/4 = Sum_{n >= 1} i/(n*P(n, i)*P(n-1, i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n*A006139(n)*A006139(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(3))^n ). - Peter Bala, Mar 16 2024
Equals arctan( phi^(-3) ) + arctan(phi^(-1) ). - Gary W. Adamson, Mar 27 2024
Equals Sum_{n>=1} eta(n)/2^n, where eta(n) is the Dirichlet eta function. - Antonio Graciá Llorente, Oct 04 2024
Equals Product_{k>=2} ((k + 1)^(k*(2*k + 1))*(k - 1)^(k*(2*k - 1)))/k^(4*k^2). - Antonio Graciá Llorente, Apr 12 2025
Equals Integral_{x=sqrt(2)..oo} dx/(x*sqrt(x^2 - 1)). - Kritsada Moomuang, May 29 2025
Extensions
a(98) and a(99) corrected by Reinhard Zumkeller, Nov 20 2012
Comments