OFFSET
0,1
COMMENTS
The two-parameter arithmetic-geometric mean function AGM is defined by taking the limit of the sequence of iterates of the map (x, y) -> ((x+y)/2, sqrt(x*y)). This can be extended to an arbitrary finite sequence of numbers by defining AGM(x(1), ..., x(n)) = AGM((x(1)+...+x(n))/n, (x(1)*...*x(n))^(1/n)). Different extensions of the definition to more than two parameters are also possible, such as the one used in A332093.
FORMULA
Equals Pi/4 * (1/2 + 1/e) / K(((e-2)/(e+2))^2) where K is the complete elliptic integral of the first kind.
EXAMPLE
0.431407125466772950330229198641630937300926634224766278636544...
MATHEMATICA
RealDigits[Pi/4 * (1/2 + 1/E) / EllipticK[((E-2)/(E+2))^2], 10, 100][[1]]
RealDigits[ArithmeticGeometricMean[(2 + E)/(4 E), 1/Sqrt[2 E]], 10, 100][[1]] (* Jan Mangaldan, Dec 07 2021 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Ben Whitmore, Mar 22 2021
STATUS
approved