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### 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

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

- Doctor Korsak, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Analysis
College Calculus
High School Analysis
High School Calculus
High School Sequences, Series

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