A342813 Decimal expansion of the limit of AGM(1, 2, ..., n)/n.

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

Offset: 0

Author

Ben Whitmore, Mar 22 2021

Keywords

• nonn
• cons

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.

Examples

			0.431407125466772950330229198641630937300926634224766278636544...
		

Crossrefs

Cf. A332093.

Programs

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 *)

PARI

Pi/4 * (exp(-1)+.5) / ellK(1-4/(exp(1)+2)) \\ Charles R Greathouse IV, Feb 05 2025

PARI

agm(exp(-1)/2+1/4, exp(-1/2)/sqrt(2)) \\ Charles R Greathouse IV, Feb 05 2025

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.