Convergence of $sqrt{1-sqrt{{1}-sqrt{{1}-sqrt{{1-sqrt{{1}}}...}}}}$

Question

Recently, as is evident from many of my recent questions, I have been very interested in nested radicals. Recently I attempted to investigate the following infinite nested radical and arrived at a strange, counter-intuitive result.
$$sqrt{1-sqrt{{1}-sqrt{{1}-sqrt{{1-sqrt{{1...}}}}}}}$$
At first glance, it would seem that the value of the nested radical must be either $0$ or $1$. However, if we make the assumption that a single value can be assigned to it, ie it converges, we arrive at a different value for it:
Let
$$x=sqrt{1-sqrt{{1}-sqrt{{1}-sqrt{{1-sqrt{{1...}}}}}}}implies x=sqrt{1-x}implies x^2=1-x implies x^2+x-1=0$$
Using the quadratic formula, we get $$x=frac{-1+sqrt5}{2}$$
as $x$ is necessarily positive. Can this result be true? Or is my assumption that allowed me to label the radical as $x$ and assume it converges incorrect?

miracle173 · Answer

Whenever we find $ldots$ in a formula we must be clear what they are meaning. My interpreation of a formula like
$$sqrt{1-sqrt{1-sqrt{1-sqrt{1-ldots}}}}$$
is that this is the limit of the sequence
$$x_1=sqrt{1}\
x_2=sqrt{1-sqrt{1}}\
x_3=sqrt{1-sqrt{1-sqrt{1}}}\
ldots\
x_{n+1}=sqrt{1-x_n}$$
If this is the meaning of this expression then we have
$$x_1=1\
x_2=0\
x_3=1\
x_4=0\
x_5=1\
x_6=0\
ldots
$$
and this sequence does not converge at all.

Your calculation says:

If $x_n$ converges, then the limit will be $frac{-1+sqrt5}{2}$ or $frac{-1-sqrt5}{2}$

This is true. But $x_n$ does not converge, so this result is useless.

Convergence of $sqrt{1-sqrt{{1}-sqrt{{1}-sqrt{{1-sqrt{{1}}}...}}}}$

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