How do we find the general term from an infinite nested radical

Question

I am struggling on how we definite infinite nested radical.

Consider $ {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots }}}}}}}}$

From what I have researched, I know we can consider the above expression as the limit of a certain squence. I know how to find this value. However, I would like to find the general term of the corresponding sequence but I do not know where to start. I tried and error but failed.

Is there any formula/theorem that are specialised for finding the general terms of such thing?

Many thanks.

Answer 1

If you google for the value of trigonometric functions of angles $\theta_n=\frac \pi {2^n}$ (for $n=2,3,4,5$, you will find them in http://mathworld.wolfram.com, you will notice $$\cos\left(\frac \pi {4}\right)=\frac 12 \sqrt 2$$ $$\cos\left(\frac \pi {8}\right)=\frac 12 \sqrt{ 2+\sqrt 2}$$ $$\cos\left(\frac \pi {16}\right)=\frac 12 \sqrt{ 2+{\sqrt {2+\sqrt 2}}}$$ $$\cos\left(\frac \pi {32}\right)=\frac 12 \sqrt{ 2+{\sqrt {2+{\sqrt {2+\sqrt 2}}}}}$$

Do you see the pattern ?

Answer 2

Hint: Set $x=\sqrt{2+\sqrt{2\ldots}}$. Then $x^2 = 2 + x$.