Possible Duplicate:
Functional Iterations (numerical analysis)
Let F be a function $F:\mathbb R\to\mathbb R$ and Let $x_0$ be (any) real number. we define : $x_{n+1}=F(x_n)\, (\text{ for }n\ge 0)$
The book I'm reading on numerical analysis claims that the order of convergence of $\{x_i\}$ ($i$ goes from $0$ to $\infty$) is the first integer q such that the q-th derivative of F is not zero, but this claim is left as an exercise.
I need some help proving this claim, I thought about Taylor but I'm not sure how can it help...
edit: I think we also need to assume that F is a contractive map
another edit: : I have found this claim in another book:
Assume that $φ(x)$ is $p$ times continuously differentiable. Then the iteration method $x_{n+1} = φ(x_n)$ is of order $p$ for the root $α$ if and only if $φ(j)(α) = 0, j = 1 : p − 1, φ(p)(α) != 0$.