A019669 Decimal expansion of Pi/2.

1, 5, 7, 0, 7, 9, 6, 3, 2, 6, 7, 9, 4, 8, 9, 6, 6, 1, 9, 2, 3, 1, 3, 2, 1, 6, 9, 1, 6, 3, 9, 7, 5, 1, 4, 4, 2, 0, 9, 8, 5, 8, 4, 6, 9, 9, 6, 8, 7, 5, 5, 2, 9, 1, 0, 4, 8, 7, 4, 7, 2, 2, 9, 6, 1, 5, 3, 9, 0, 8, 2, 0, 3, 1, 4, 3, 1, 0, 4, 4, 9, 9, 3, 1, 4, 0, 1, 7, 4, 1, 2, 6, 7, 1, 0, 5, 8, 5, 3

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

With offset 2, decimal expansion of 5*Pi. - Omar E. Pol, Oct 03 2013

Decimal expansion of the number of radians in a quadrant. - John W. Nicholson, Oct 07 2013

Not the same as A085679. First differing term occurs at 10^-49, as list -49, or 51st counting term (a(-49)= 5 and A085679(-49) = 4). - John W. Nicholson, Oct 07 2013

5*Pi is also the surface area of a sphere whose diameter equals the square root of 5. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013

Pi/2 is also the radius of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Dec 27 2013

Examples

			Pi/2 = 1.570796326794896619231321691639751442098584699...
5*Pi = 15.70796326794896619231321691639751442098584699...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.4.1 and 1.4.2, pp. 20-21.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, p. 185 (preprint draft).
• Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.
• Michael Penn, A nice sum from the Harvard MIT math trust, YouTube video, 2022.
• L. D. Servi, Nested Square Roots of 2, The American Mathematical Monthly 110:4 (Apr. 2003), pp. 326-330.
• Johan Wästlund, An Elementary Proof of the Wallis Product Formula for pi, The American Mathematical Monthly 114:10 (Dec. 2007), pp. 914-917.
• Eric W. Weisstein and Jonathan Sondow, Wallis Formula, MathWorld.
• Wikipedia, Viète's formula.
• Index entries for transcendental numbers.

Crossrefs

Cf. A053300 (continued fraction), A060294 (2/Pi).

Cf. A000796, A019692, A122952, A019694 (Pi through 4*Pi), A106854.

Programs

Maple

Digits:=100: evalf(Pi/2); # Wesley Ivan Hurt, Oct 26 2016

Mathematica

RealDigits[N[Pi/2, 200]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

PARI

default(realprecision, 20080); x=Pi/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019669.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

Formula

Pi/2 = log(i)/i, where i = sqrt(-1). - Eric Desbiaux, Jun 27 2009

Pi/2 = Product_{n>=1} (n/(n+1))^((-1)^n) = 2 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * 8/7 * 8/9 * 10/9 * ... (Wallis formula). - William Keith and Alonso del Arte, Jun 24 2012

Equals Sum_{k>1} 2^k/binomial(2*k,k). - Bruno Berselli, Sep 11 2015

The previous result is the particular case n = 1 of the more general identity: Pi/2 = 4^(n-1) * n!/(2*n)! * Sum_{k >= 2} 2^(k+1)*(k + n - 1)!*(k + 2*n - 2)!/(2*k + 2*n - 2)! valid for n = 0,1,2,... . - Peter Bala, Oct 26 2016

Pi/2 = Product_{n>=1} (4*n^2)/(4*n^2-1). - Fred Daniel Kline, Oct 29 2016

Pi/2 = lim_{n->oo} F(2^(n+3))/2, with one half of the area of a regular 2^(n+3)-gon, for n >= 0, inscribed in the unit circle, written as iterated square roots of 2 as F(2^(n+3))/2 = 2^n*sqrt(2 + sq2(n)), with sq2(n) = sqrt(2 + sq2(n-1)), n >= 1, with input sq2(0) = 0 (2 appears n times in sq2(n)). Viète's infinite product formula works with the partial product F(2^(n+2))/2 = Product_{j=1..n} (2/sq2(j)), n >= 1, which corresponds to the above given formula. - Wolfdieter Lang, Jul 06 2018

Pi/2 = Integral_{x = 0..oo} sin(x)^2/x^2 dx = 1/2 + Sum_{n >= 1} sin(n)^2/n^2, by the Abel-Plana formula. - Peter Bala, Nov 05 2019

From Amiram Eldar, Aug 15 2020: (Start)

Equals Sum_{k>=0} k!/(2*k + 1)!!.

Equals Sum_{k>=0} (-1)^k/(k + 1/2).

Equals Integral_{x=0..oo} 1/(x^2 + 1) dx.

Equals Integral_{x=0..oo} sin(x)/x dx.

Equals Integral_{x=0..oo} exp(x/2)/(exp(x) + 1) dx.

Equals Product_{p prime > 2} p/(p + (-1)^((p-1)/2)). (End)

Pi/2 = Integral_{x = 0..oo} 1/(1 - x^2 + x^4) dx = (1 + 2/3 + 1/5) - (1/7 + 2/9 + 1/11) + (1/13 + 2/15 + 1/17) - .... - Peter Bala, Jul 22 2022

Equals arcsin(9/10) + sqrt(19)*Sum_{k >= 1} A106854(k-1)/(k*10^k) (see Bailey and Crandall, 2001). - Paolo Xausa, Jul 15 2024

Equals 2F1(1/2,1/2 ; 3/2; 1). - R. J. Mathar, Aug 20 2024

Pi/2 = [1;1,1/2,1/3,...,1/n,...] by Wallis's approximation. - Thomas Ordowski, Oct 19 2024

From Stefano Spezia, Oct 21 2024: (Start)

Equals Sum_{k>=0} 2^k/((2*k + 1)*binomial(2*k,k)) (see Finch).

Equals Limit_{n->oo} 2^(4*n)/((2*n + 1)*binomial(2*n,n)^2) (see Finch). (End)

Equals Integral_{x=-oo..oo} sech((2*x^3 + x^2 - 5*x)/(x^2 - 1)) dx. - Kritsada Moomuang, May 29 2025