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Some continued fractions converge very fast. Example

However, the convergence of the famous continued fraction of $\sqrt{2}$,

$$a_1 = 1, a_{n + 1} = 1 + \frac{1}{1 + a_n}$$

is linear.

$$|a_{n+1}-\sqrt{2}|=\frac{(\sqrt{2}-1)|a_n-\sqrt{2}|}{1+a_n}$$

Is there a continued fraction that converges to $\sqrt{x}$ at a quadratic or faster rate?

Gary
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    The exposition at the link does not limit itself to simple continued fractions, that is, continued fractions where the partial numerators are all $1$; it allows more general continued fractions, where arbitrary partial numerators are allowed. Maybe that would allow one to find faster convergence to $\sqrt x$, I don't know. A limiting factor would be that for each positive nonsquare integer $n$ there is a positive constant $c$ such that $|\sqrt n-(p/q)|>cq^{-2}$ for all integer $p,q$. – Gerry Myerson Apr 19 '24 at 04:43

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