Fraction mapped to sum and geometric mean approaches 2.1479

Question

Map a positive fraction $\frac{a}{b}$ to $\frac{a + b}{\sqrt{ab}}$. Repeating seems to map every starting fraction to a number close to $\xi = 2.1479$: $$ \begin{array}{ccccc} \frac{1}{4} & \frac{5}{2} & \frac{7}{\sqrt{10}} & \frac{7+\sqrt{10}}{\sqrt{7} \sqrt[4]{10}} & \frac{7+\sqrt{7} \sqrt[4]{10}+\sqrt{10}}{\sqrt[4]{7} \sqrt[8]{10} \sqrt{7+\sqrt{10}}} \\ 0.25 & 2.5 & 2.21359 & 2.15994 & 2.1501 \\ \end{array} $$

Q. Does indeed every starting fraction approach $\xi\,$? Is $\xi$ a known constant in other contexts?

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This is a variation on an earlier post, Why does this process map every fraction to the golden ratio?

Answer 1

$${a+b\over\sqrt{ab}}={(a/b)+1\over\sqrt{a/b}}$$ so you are mapping $x$ to $(x+1)/\sqrt x=x^{1/2}+x^{-1/2}$. Fixed point should be solution of $x=x^{1/2}+x^{-1/2}$. This is a cubic in $x$, $x^3-x^2-2x-1=0$.