TITLE: The value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$?
QUESTION [11 upvotes]: How to find value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$ ? 
I've calculated it by MATLAB for some finite terms and I've got : $0.3001 - 0.4201i$, but I don't know how to find the value analytically! Would you mind helping me find it? Thanks

REPLY [3 votes]: This question was already answered and one answer was accepted. This answer is just to add to the completeness for some later reader.     
If one looks at the function $f(x) = \sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+x}}}}$ then this function has true fixpoints; the fixpoints are $\small t_0 = 0$, $ \small t_1 \approx 0.222985944830 + 0.413363796251 i$ and $ \small t_2 \approx 0.222985944830 - 0.413363796251 i$.     
Around the fixpoint $t_0$ we get a real power series with a fractional cofaktor but without constant term due to wolframalpha, see my related question.     
Developing around the complex fixpoints, say $t_1$ we get a usual (though with complex coefficients) power series, $ \small f_4(x+t_1)-t_1 = g(x) \approx 0.219471696356 x + $  $ \small (-0.102599142965 + 0.177579081091 i) x^2 + $ $ \small(-0.133743852861 - 0.220457785139 i) x^3 $  $ \small+ (0.375580757794 - 0.0217915275445 i) x^4 + O(x^5) $ where the absolute value of the coefficient of the linear term is smaller than 1 - which means, that this fixpoint $t_1$ is also attracting.       
Thus infinite iteration of $f(x)$ beginning from any complex value (except 0) shall converge to that fixpoint $t_1$. This solves the first problem: is that OP's notation converging at all.
After that we can simply state, that any finite truncation of the OP's expression approximates one of the cycling fixpoints :    
t1=0.222985944830 + 0.413363796251*I
x0=sqrt(1+t1)     :    %2164 = 1.1211469 + 0.18434863*I
x0=sqrt(1-x0)     :    %2165 = 0.22298594 - 0.41336380*I
x0=sqrt(1+x0)     :    %2166 = 1.1211469 - 0.18434863*I
x0=sqrt(1-x0)     :    %2167 = 0.22298594 + 0.41336380*I  \\ cycling occurs, approxi-
x0=sqrt(1+x0)     :    %2168 = 1.1211469 + 0.18434863*I   \\ mating the four-point
x0= ...           :     ...  =  ...                       \\  cycle