Please any help to Compute the following limit

Question

Let $-1<a_0<1$ and define recursively $a_n=((1+a_{n-1})/2)^{(1/2)}$. If $A_n=4^n(1-a_n)$ compute $\lim_{n \to \infty} A_n$.

I have showed that ${a_n}$ converges to $1$ and that $A_n$ is increasing. But i can't to show that $A_n$ is bounded or something that allow to concluide the convergence of $A_n$. Please any help or suggestion. Thanks.

Answer 1

let $a_0 = \cos \theta\;$ then $$ a_1 = \sqrt{\frac{1+\cos \theta}2} = \cos \frac{\theta}2 $$ so $$ a_n = \cos \frac{\theta}{2^n} $$ and $$ 1-a_n =2 \sin^2 \frac{\theta}{2^{n+1}} $$ perhaps you can take this a little further...