# Fraction mapped to sum and geometric mean approaches 2.1479

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?

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

• twice arithmetic mean over geometric mean. – Roddy MacPhee Nov 19 '19 at 0:14
• That's surely not correct. Doing this with the OP's example would give $\frac{11}{2 \sqrt10} \approx 1.74$ (Wolfram Alpha). – Toby Mak Nov 19 '19 at 0:20

$${a+b\over\sqrt{ab}}={(a/b)+1\over\sqrt{a/b}}$$ 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 $$x$$, $$x^3-x^2-2x-1=0$$.
• Nice! The sole real root of that cubic $\approx 2.1478990$. – Joseph O'Rourke Nov 19 '19 at 0:39