Let $f(x)=\sqrt{1-x^2}$, $b = 1/\sqrt{2}$. The sequence $(E_n)_{n=1}^{\infty}$ is defined as the solution to the following equation :
$$f(E_n - f(E_n -f(E_n - ....-f(E_n - b)))) = E_n -1,$$
where the function composition is performed $n$ times. So for example :
$$f(E_1 -b) = \sqrt{1 - (E_1-b)^2} = E_1 - 1 \Rightarrow E_1 \simeq 1.545,$$ $$f(E_2 - f(E_2 -b)) = \sqrt{1 - \left(E_2-\sqrt{1 - (E_2-b)^2}\right)^2} = E_2 - 1 \Rightarrow E_2 \simeq 1.491,$$ $$f(E_3-f(E_3 - f(E_3 -b))) = \sqrt{1 - \left(E_3-\sqrt{1 - \left(E_3-\sqrt{1 - \left(E_3-b\right)^2}\right)^2}\right)^2} = E_3 - 1 \Rightarrow E_3 \simeq 1.466.$$
We also have $E_4 \simeq 1.451$ and $E_5 \simeq 1.442$.
The problem arises in a system of equations with several variables and using some other arguments I can prove that the sequence $E_n$ is decreasing and converges to $E_{\infty} := 2\cos(\pi/4) = \sqrt{2} \simeq 1.414$. I will be happy to provide more details if it can be useful. However I figured I would try to provide a minimal amount of information.
My question is, what is the rate of convergence of $E_n$ ? Numerically I would guess for $E_n - E_{\infty} = O(1/n^2)$.
The graph below illustrates the solutions $E_n$ for $n=1,2,3,4,5$ as the intersections of the functions $f(x-f(x-....-f(x-b))))$ and the dotted straight line $x-1$. The points $N_1$, $N_2$ and $N_3$ appear to be common points of intersections of the nested functions. The $N_3$ point has $x$-coordinate $\sqrt{2}$ which is $E_{\infty}$.
Any ideas or references or straight up proofs are very welcome. Thanks
