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===Limits of the arithmetic-geometric mean===
The [[arithmetic-geometric mean]] of two numbers, <math>a_0\!</math> and <math>b_0\!</math>, is found by calculating the limit of the sequences <math>a_{n+1}={a_n+b_n \over 2}\!</math>, <math>b_{n+1}=\sqrt{a_n b_n}\!</math>, which both converge to the same limit.
If <math>a_0=1\!</math> and <math>b_0=\cos\phi\!</math> then the limit is <math>{\pi \over 2K(\sin\phi)}\!</math> where <math>K(k)\!</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]]
:<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}\!</math>.
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