I have been playing around with rational approximations to square roots. I am familiar with https://en.wikipedia.org/wiki/Methods_of_computing_square_roots but did not see it there.
I also did not see the formula on https://en.wikipedia.org/wiki/Square_root_of_2#Series_and_product .
I ended up with this formulation and I wondered whether it was known. \begin{align} \sqrt{a} = \lim_{n\to \infty}\frac{\sum_{i=0, i \textrm{ even}}^n {n \choose i} x^{n-i} a^{i/2}}{\sum_{i=0, i \textrm{ odd}}^n {n \choose i} x^{n-i} a^{(i-1)/2}} \end{align} for all $x$. In particular, for $x=1$, we have \begin{align} \sqrt{a} = \lim_{n\to \infty}\frac{\sum_{i=0, i \textrm{ even}}^n {n \choose i} a^{i/2}}{\sum_{i=0, i \textrm{ odd}}^n {n \choose i} a^{(i-1)/2}} \; . \end{align} The convergence is linear.
I assume this is known but could not find a reference.
Thanks for your help.