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I came across the following interesting exercise:

Let $b_{n+1}=\frac{1}{2}(a_n+b_n)$ and $a_{n+1}=\sqrt{a_nb_n}$ with $a_1>0$, $b_1>0$ and $b_1>a_1$. Show that $I_n=[a_n, b_n]$ is a sequence of nested intervals.

My question is not about the fact that $I_n$ is a sequence of nested intervals. This is relatively straightforward to show. But I have not been able to calculate the limiting value $\lim\limits_{n\to\infty} a_n = \lim\limits_{n\to\infty} b_n$ as a function of $a_1$ and $b_1$ (even though this is not part of the exercise). Any ideas are greatly appreciated!

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marked as duplicate by J. M., Stefan Hansen, Brian M. Scott, Asaf Karagila, vonbrand Apr 3 '13 at 8:41

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up vote 1 down vote accepted

The limit is called the arithmetic-geometric mean $M(a_1,b_1)$. An integral-form expression is $$ M(x,y) = \frac{\pi}{4}\cdot\frac{x + y}{K\left( \frac{x - y}{x + y} \right)}, $$ where $K$ is the complete elliptic integral of the first kind.

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Thanks for your fast reply! I'm glad you pointed the arithmetic-geometric mean out to me! –  Matt L. Apr 2 '13 at 20:04
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