Infinite limit terms under root Suggest me a hint to solve:$$\psi=\lim_{x\to0}{\frac{\sqrt{1-\cos x+\sqrt{1-\cos x+\sqrt{1-\cos x+\cdots}}}-1}{x^2}}$$

My try, 


Hint
If you use Taylor series, you can start from lab bhattacharjee's answer
$$y=\frac{1\pm\sqrt{5-4\cos x}}2$$ and use the fact that around $x=0$, $\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}+O\left(x^5\right)$. So,$$5-4\cos (x)=1+2 x^2-\frac{x^4}{6}+O\left(x^5\right)$$ Now, use the fact that $$\sqrt{1+y}=1+\frac{y}{2}-\frac{y^2}{8}+O\left(y^3\right)$$ and replace $y$ by $(2 x^2-\frac{x^4}{6})$. So,$$\sqrt{5-4\cos (x)}=1+x^2-\frac{7 x^4}{12}+O\left(x^5\right)$$
I am sure that you can take from here and easily find lab bhattacharjee's results.
