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I showed that the limit exists and that $\lim\limits_{n\to \infty }a_n=\lim\limits_{n\to \infty }b_n$. My question is how to compute the limit assuming $a_1$ and $a_2$ are given positive numbers.

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The Wiki page on arithmetic-geometric mean contains an explicit answer together with its derivation. The formula for the limit is rather nontrivial and involves complete elliptic integrals of the first kind. Namely, $$M(a_1,a_2)=\frac{\pi}{4}\frac{a_1+a_2}{K\left(\frac{a_1-a_2}{a_1+a_2}\right)}.$$

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  • $\begingroup$ Accordingly, Gauss proposed that recurrence as an efficient way to compute $K(z)$. $\endgroup$ – GEdgar May 3 '15 at 12:42

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