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%I #104 Oct 08 2022 09:42:29
%S 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,
%T 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,
%U 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
%N Decimal expansion of Pi/2.
%C With offset 2, decimal expansion of 5*Pi. - _Omar E. Pol_, Oct 03 2013
%C Decimal expansion of the number of radians in a quadrant. - _John W. Nicholson_, Oct 07 2013
%C 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
%C 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
%C 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
%H Harry J. Smith, <a href="/A019669/b019669.txt">Table of n, a(n) for n = 1..20000</a>
%H Michael Penn, <a href="https://www.youtube.com/watch?v=PjrXHxaIBDo">A nice sum from the Harvard MIT math trust</a>, YouTube video, 2022.
%H L. D. Servi, <a href="https://www.jstor.org/stable/3647881">Nested Square Roots of 2</a>, The American Mathematical Monthly 110:4 (Apr. 2003), pp. 326-330.
%H Johan Wästlund, <a href="http://www.math.chalmers.se/~wastlund/monthly.pdf">An Elementary Proof of the Wallis Product Formula for pi</a>, The American Mathematical Monthly 114:10 (Dec. 2007), pp. 914-917.
%H Eric W. Weisstein and Jonathan Sondow, <a href="http://mathworld.wolfram.com/WallisFormula.html">Wallis Formula</a>, MathWorld.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula">Viète's formula</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Pi/2 = log(i)/i, where i = sqrt(-1). - _Eric Desbiaux_, Jun 27 2009
%F 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
%F Equals Sum_{k>1} 2^k/binomial(2*k,k). - _Bruno Berselli_, Sep 11 2015
%F 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
%F Pi/2 = Product_{n>=1} (4*n^2)/(4*n^2-1). - _Fred Daniel Kline_, Oct 29 2016
%F 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
%F 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
%F From _Amiram Eldar_, Aug 15 2020: (Start)
%F Equals Sum_{k>=0} k!/(2*k + 1)!!.
%F Equals Sum_{k>=0} (-1)^k/(k + 1/2).
%F Equals Integral_{x=0..oo} 1/(x^2 + 1) dx.
%F Equals Integral_{x=0..oo} sin(x)/x dx.
%F Equals Integral_{x=0..oo} exp(x/2)/(exp(x) + 1) dx.
%F Equals Product_{p prime > 2} p/(p + (-1)^((p-1)/2)). (End)
%F 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
%e Pi/2 = 1.570796326794896619231321691639751442098584699...
%e 5*Pi = 15.70796326794896619231321691639751442098584699...
%p Digits:=100: evalf(Pi/2); # _Wesley Ivan Hurt_, Oct 26 2016
%t RealDigits[N[Pi/2, 200]] (* _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009 *)
%o (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
%Y Cf. A053300 (continued fraction), A060294 (2/Pi).
%Y Cf. A000796, A019692, A122952, A019694 (Pi through 4*Pi).
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_
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