We consider $I$ a real interval of strictly positive length, $f$ a function defined on $I$ with values in $I$ and $\left(u_{n}\right)_{n \in \mathbb{N}}$ a sequence defined by $u_{0} \in I$ and $\forall n \in \mathbb{N}, u_{n+1}=f\left(u_{n}\right)$. We assume that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ converges to an element $\ell$ of $I$ and that $f$ is differentiable at $\ell$. a) Show that $f(\ell)=\ell$. b) Show that if the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary then it belongs to $E^{c}$. Give its convergence rate as a function of $f^{\prime}(\ell)$. c) Show that if $\left|f^{\prime}(\ell)\right|>1$, then $\left(u_{n}\right)_{n \in \mathbb{N}}$ is stationary. d) Let $r$ be an integer greater than or equal to 2. We assume that the function $f$ is of class $\mathcal{C}^{r}$ on $I$ and that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary. Show that the convergence rate of $\left(u_{n}\right)_{n \in \mathbb{N}}$ is of order $r$ if and only if $\forall k \in\{1,2, \ldots, r-1\}, f^{(k)}(\ell)=0$.
We consider $I$ a real interval of strictly positive length, $f$ a function defined on $I$ with values in $I$ and $\left(u_{n}\right)_{n \in \mathbb{N}}$ a sequence defined by $u_{0} \in I$ and $\forall n \in \mathbb{N}, u_{n+1}=f\left(u_{n}\right)$. We assume that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ converges to an element $\ell$ of $I$ and that $f$ is differentiable at $\ell$.
a) Show that $f(\ell)=\ell$.
b) Show that if the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary then it belongs to $E^{c}$. Give its convergence rate as a function of $f^{\prime}(\ell)$.
c) Show that if $\left|f^{\prime}(\ell)\right|>1$, then $\left(u_{n}\right)_{n \in \mathbb{N}}$ is stationary.
d) Let $r$ be an integer greater than or equal to 2. We assume that the function $f$ is of class $\mathcal{C}^{r}$ on $I$ and that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary. Show that the convergence rate of $\left(u_{n}\right)_{n \in \mathbb{N}}$ is of order $r$ if and only if $\forall k \in\{1,2, \ldots, r-1\}, f^{(k)}(\ell)=0$.