Infinite Series Involving Arithmetic and Geometric Means

Date: 11/27/2003 at 00:41:13
From: Nikolov 
Subject: An Infinite Series Involving Arithmetic and Geometric Means

I am curious whether or not the following series would converge or 
diverge, and if it converges, to what value? Actually, I'm not sure 
if this is even a series!  I'm not sure how to put this symbolically, 
so let me explain the series.

Take a set of numbers (let us use the two numbers {a, b}), and take
both the arithmetic and geometric mean of the set:

  A = (a+b)/2   [arithmetic mean, A]
  G = sqrt(ab)  [geometric mean, G]

Now, take the arithmetic and geometric means of those two results:

  A2 = {[(a+b)/2]+[sqrt(ab)}/2
  G2 = sqrt{[(a+b)/2]*[sqrt(ab)]}

Now, do the same thing over again with those results, and keep doing
it over and over again ad infinitum.  What would the result be as the
number of iterations of this process approaches infinity?

Upon closer examination, it looks like it may be indefinite due to 
the fact that it may oscillate between the values of A(n) and G(n). 
However, when I plug in values for a and b, the series clearly 
converges:

{a, b} = {21, 53}

A(1) = 37        G(1) = 33.362
A(2) = 35.1841   G(2) = 35.134
A(3) = 35.1575   G(3) = 34.654
A(4) = 34.906    G(4) = 34.905
A(5) = 34.905    G(5) = 34.905

{a, b} = {1/2, 2/3}

A(1) = .5833        G(1) = .577
A(2) = .5803        G(2) = .5803
A(3) = .5803        G(3) = .5803

On the infinite scale, however, this series may do strange things. 
With some kind of formula, it may be easy to answer this question.  

Also, while on the subject of means, what other kinds of means are 
there besides geometric and arithmetic means?

Thanks!

--Nikolov

Date: 11/27/2003 at 04:37:29
From: Doctor Korsak
Subject: Re: An Infinite Series Involving Arithmetic and Geometric Means

Hello Nikolov,

You posed a most interesting problem!  I finally gave up on it and 
searched the web, and found the solution for you at

  Arithmetic-Geometric Mean
    http://mathworld.wolfram.com/Arithmetic-GeometricMean.html 

Do you know about the proof of the so-called "Fermat's Last Theorem"?  
This problem has a connection, due to its involvement with "elliptic 
curves", as you will see at the above URL.

There is also a fascinating connection of your problem to a practical 
calculation of arctan(x).  You can read about it here:

  Inverse Tangent
    http://mathworld.wolfram.com/InverseTangent.html 

where the iteration is:

  a_0 = sqrt(1+x^2),  b_0 = 1,

  a_i+1 = (a_i + b_i)/2  , b_i+1 = sqrt(a_i+1 * b_i), 

almost the same as your double progression, and then

  arctan(x) = limit as n--> infinity of(x / (a_n * sqrt(1+x^2))) .

As for the last part of your question, there is also the "harmonic 
progression".  You can find a discussion of it at
 
  Harmonic Mean 
    http://mathforum.org/library/drmath/view/57565.html 

Please contact Dr. Math if you need further help.

- Doctor Korsak, The Math Forum
  http://mathforum.org/dr.math/