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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 $φ(j)(α) = 0, j = 1 : p − 1, φ(p)(α) != 0$.

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    The statement of the question is a little bit unsatisfactory, I have to say. You already correctly added an assumption which guarantees convergence (which could be improved), but still the question remains what exactly the hypotheses on the derivatives is. If all derivatives of $F$ are (identically) zero up to some order $q>0$, then $F =0$. This is already true in case of $q=1$. So you need to state at which point(s) you are making a hypothesis about the derivatives of $F$. And yes, Taylor is very likely one key to such an exercise.2011-12-17
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    @Thomas - found this in another book (I still can't prove this though...)2011-12-17

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