Assume that the fixed point iteration for computing the fixed point $p=0$ of the function $g(x) = cos(x^{50})-1$ converges. Determine the order of convergence.
We know the order of convergence is $\alpha$ if $\lim_{n\to\infty} \frac{|({p_{n+1}-p})|}{|p_n-p|^\alpha} = \lambda$ for some positive $\alpha$ and $\lambda$.
By applying Taylor's expansion, we obtain $g(x) = g(p) + (x-p)g(p) + \cdots+ \frac{(x-p)^n}{n!}g^{(n)}(p) + \cdots$
And applying it to the above definition, I guess that the order of convergence is 50. This is observed through differentiating the function ~50 times and finding that $g^{(50)}(p)$ is the first time $g^{(n)}(p) \neq 0$, but I have no idea on how to formalize this. Any help will be appreciated
Near $x=0$ you have: $$\cos(x^{50})-1=\left(1-\frac{(x^{50})^2}{2}+o(x^{100}) \right)-1=-\frac{x^{100}}{2}+o(x^{100})$$ so if $p_n \to 0$ then: $$g(p_n) = -\frac{p_n^{100}}{2}+o(p_n^{100})$$ and so with $\alpha=100$ you have: $$\frac{|g(p_n)-0|}{|p_n-0|^{100}}=-\frac{1}{2}+o(1)$$ so the order of convergence to $0$ is $100$.