We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th iteration.

This leads to the question what classes of continued fractions can be given as Arithmetic geometric procedure?

1. Does $C(x) = x + \frac{1^{2}}{2x + \frac{3^{2}}{2x + \frac{5^{2}}{2x + \frac{7^{2}}{2x + \cdots}}}}$ have a rapid Arithmetic geometric procedure at $x\in\Bbb Z_{\neq1}$ (note $C(1)=\frac{4}{\pi}$)?

2. Which algebraic numbers that have continued fraction expansion, admit rapid Arithmetic geometric procedure?

3. Since $\pi$ has rapid Arithmetic geometric procedure we have $\zeta(2n)$ having rapid Arithmetic geometric procedure. Does $\zeta(2n+1)$ admit rapid Arithmetic geometric procedure?

4. Supposing the fastest approximation for a number is through a rapid Arithmetic geometric procedure, is that number transcendental? Is there any connection to transcendentality through existence of rapid Arithmetic geometric procedure?