Infinite Series Involving Arithmetic and Geometric MeansDate: 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/ |
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