OFFSET
0,1
COMMENTS
Probability that a point selected uniformly at random from the unit 4-cube is in the unit 4-sphere.
Let S(n) = 1 - 1/3 + 1/5 - ... + ((-1)^(n-1))/(2n-1). Then Sum{n >=1} ((-1)^(n-1))*S(n) /(2n+1) = Pi^2 /32. The convergence is very slow. - Michel Lagneau, Feb 27 2015
FORMULA
Equals Integral_{0..infinity} x^2*BesselK(0, x)^2 dx. - Jean-François Alcover, Apr 15 2015
Equals Integral_{x=0..1} arctan(x)/(1+x^2) dx. - Amiram Eldar, Aug 09 2020
Equals Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} (1 + x^2 + y^2 + z^2)^(-2). - Peter Luschny, Dec 10 2022
EXAMPLE
Choose -1 <= w, x, y, z <= 1 uniformly at random. Then this constant is the probability that w^2 + x^2 + y^2 + z^2 <= 1.
MAPLE
Digits:=100; evalf(Pi^2/32); # Wesley Ivan Hurt, Feb 27 2015
MATHEMATICA
RealDigits[Pi^2/32, 10, 120][[1]] (* Harvey P. Dale, Jul 13 2014 *)
PROG
(PARI) Pi^2/32
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Charles R Greathouse IV, Jul 07 2014
STATUS
approved