Evaluating $\sqrt{1+\sqrt{2 - \sqrt{3 + \sqrt{4 - \cdots}}}}$ 
$$x =\sqrt{1+\sqrt{2 - \sqrt{3 + \sqrt{4 - \sqrt{5+ \sqrt{6 - \cdots}}}}}}$$
Find $x$.

I am not sure how to proceed.
Is this a sort of Arithmetico-Geometric Progression? Will this converge at a point?
Any help would be sincerely appreciated.


This is just an empirical answer. Consider
$$f(n) = \sqrt{t+\sqrt{t+1 - \sqrt{t+2 + \sqrt{t+3 - \cdots}}}}$$
We need to find $f(1)$. But
\begin{align}
f(1) &= \sqrt{1+\sqrt{2 - f(3)}}\\
f(3) &= \sqrt{3+\sqrt{4 - f(5)}}\tag{Similarly moving on }\\
f(n) &= \sqrt{n+\sqrt{n+1 - f(n+2)}}\\
\end{align}
After squaring and rearranging, we get
$$f(n+2) = n+1 - (f(n)^2 - n)^2$$
I used the wolfram alpha recurrence solver to solve the recurrence and it gave me this:
$$\begin{array}{|c|c|c|c|}
\hline
n& 0 & 1 & 2 & 3 & 4 \\ \hline
 g(n) & 0  & 1 & 1 & 2& 2\\ \hline
\end{array}$$
I don't know if this helps, but $x = 1$.
