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## Connection between Infinite continued fractions and AGM

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?

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 Mathworld says that $C(1) = 4/\pi$, not $\pi$. – S. Carnahan♦ Dec 23 at 5:30 I meant represents in a crude way. I can correct it. – unknown (google) Dec 23 at 8:30 Okay, sorry to jump on you like that. It is a good question. – S. Carnahan♦ Dec 23 at 15:35