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The [[arithmetic–geometric mean]] of two numbers, a<sub>0</sub> and b<sub>0</sub>, is found by calculating the limit of the sequences
:<math>\begin{align} a_{n+1} & = \frac{a_n+b_n}{2}, \\[6pt]
b_{n+1} & = \sqrt{a_n b_n},
\end{align}
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:<math>K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}.\!</math>
If <math>c_0 = \sin\varphi\!</math>, <math>c_{i+1} = a_i - a_{i+1}\!</math>
:<math>\sum_{i=0}^\infty 2^{i-1} c_i^2 = 1 - {E(\sin\varphi)\over K(\sin\varphi)}\!</math>
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