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It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + \frac{3^{2}}{2x + \frac{5^{2}}{2x + \frac{7^{2}}{2x + \cdots}}}}$$

Are there any other $x$ that $C(x)$ can be approximated through AGM quickly? In general is there a comprehensive study between Infinite Continued fractions and their calculations through AGM?

Why does AGM work at $x=1$?

Is there any connection to elliptic integrals?

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Mathworld says that $C(1) = 4/\pi$, not $\pi$. – S. Carnahan Dec 23 '12 at 5:30
I meant represents in a crude way. I can correct it. – Turbo Dec 23 '12 at 8:30
Okay, sorry to jump on you like that. It is a good question. – S. Carnahan Dec 23 '12 at 15:35
The AGM doesn't make any use of that continued fraction for $4/\pi$, does it? So there's no reason to think there's any connection between AGM and $C(x)$, right? – Gerry Myerson Jun 26 '13 at 1:58

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