What happens if you repeatedly take the arithmetic mean and geometric mean?

Given two positive real numbers, $A$ and $B$, such that $A\leq B$, take the geometric mean, giving $A'$, and the arithmetic mean, giving $B'$. Repeat ad infinitum. My intuition tells me that, since both means give values between the two original numbers, they will converge as the number of repetitions approaches infinity. Is this correct? Is there a simple formula to determine on what value they converge?

Solution 1:

Indeed, they converge to what is known as the Arithmetic–geometric mean. Unfortunately there is no simple formula, but calculating it recursively, at least to a certain precision, could be considered "simple".

To see that it converges, let $a_0=A$, $b_0=0=B$ and

$$a_{n+1}=\sqrt{a_nb_n}$$ $$b_{n+1}=\frac{a_n+b_n}{2}$$

So that, by the AM-GM-inequality $$a_n\le a_{n+1} \le b_{n+1}\le b_n$$ for all n. And the difference $$b_n-a_n\le\frac{b_0-a_0}{2^n}$$ converges to zero. The reason this last inequality holds is that the difference clearly shrinks faster than if we in stead let $b_n$ be constant, in which case the difference would be divided by two at each step.

Solution 2:

This is the arithmetic-geometric mean. Yes, it does converge for positive starting values. No, there is no simple formula for the mean, although there are some interesting equivalents in the form of integrals (see the link).