Find $\lim_{n \rightarrow \infty } 2^{n} \sqrt{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}$

Question

$\lim_{n \rightarrow \infty } 2^{n} \sqrt{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}$ where the 2 inside the roots appear n times. For example if n = 2 : $2^{2} \sqrt{2-\sqrt{2}}$ I discovered this. Has this been already developed/made before?

Answer 1

Since $$\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}$$ is the side length of the regular $2^{n+1}$-gon inscribed in a radius $1$ circle, the sum of the $2^{n+1}$ sides will converge to the circumference of that circle. Thus $$\lim_{n\rightarrow\infty}2^{n+1}\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2\pi.$$