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