From: "denis.feldmann" <denis.feldmann@wanadoo.fr>
Subject: Re: limit formula asked
Date: Fri, 17 Dec 1999 17:56:07 +0100
Newsgroups: sci.math
Keywords: sequence rapidly converging to arithmetic-geometric mean

manuel a écrit dans le message <83dnt2$6gj$1@news.si.fct.unl.pt>...
>Define 2 sequences xn and yn with:
>x1:=a; y1:=b( 0<a<=b)
>x(n+1):= sqrt(xn*yn)
>y(n+1):= (xn+yn)/ 2
>
>It is easy to prove that the they converge, and I want a formula depending
>on a and b for the value of the limit of ( xn ,a ,b ).
>
>manuel

Just for the fun of being first to answer: if M(a,b) is the limit,
pi/M(a,b)= int(1/sqrt((x^2+a^2)(x^2+b^2), x= -infinity..infinity)

M(a,b) is the arithmetico-geometric mean;  the integral is an elliptic one
(no elementary expression); the result is due to Gauss (proof use the change
of variable
 y=x-ab/x (!))
==============================================================================

From: spamless@Nil.nil
Subject: Re: limit formula asked
Date: 17 Dec 1999 18:53:19 -0500
Newsgroups: sci.math

manuel <mnas@mail.fct.unl.pt> wrote:
> Define 2 sequences xn and yn with:
> x1:=a; y1:=b( 0<a<=b)
> x(n+1):= sqrt(xn*yn)
> y(n+1):= (xn+yn)/ 2

> It is easy to prove that the they converge, and I want a formula depending
> on a and b for the value of the limit of ( xn ,a ,b ).

Well, it is called the arithmetic-geometric mean of (a,b).

If a=1, b=SQRT(1-k^2) then the limit is pi/(2*K(k)) where K is the
complete elliptic integral of the first kind.
==============================================================================

From: spamless@Nil.nil
Subject: Re: limit formula asked
Date: 17 Dec 1999 18:56:37 -0500
Newsgroups: sci.math

manuel <mnas@mail.fct.unl.pt> wrote:
> Define 2 sequences xn and yn with:
> x1:=a; y1:=b( 0<a<=b)
> x(n+1):= sqrt(xn*yn)
> y(n+1):= (xn+yn)/ 2

> It is easy to prove that the they converge, and I want a formula depending
> on a and b for the value of the limit of ( xn ,a ,b ).

(note, it is easy to see that you can scale so that one of a,b is 1 and
the other is smaller than one and the result is just the limit in that
case times the scale factor ... e.g. let X=x/a, Y=y/a, b'=b/a to see that
X,Y satisfy the the same recursion with X_0=1 now)