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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1316569596882--
%%--Construction des termes d\'une suite et interprétation--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
On consid\`{e}re la fonction d\'{e}finie sur $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\backslash\{1\}$ par:%
\[
f(x)=\frac{2x+1}{x-1}%
\]
et la suite $\left(  u_{n}\right)  _{n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}}$ d\'{e}finie par:%
\[
u_{0}=\frac{1}{2}\quad\text{et}\quad\text{pour }n>0,\text{ }u_{n+1}=f(u_{n}).
\]


\begin{enumerate}
\item
\begin{enumerate}
\item \'{E}tudier les variations de $f$ sur son domaine de d\'{e}finition.

\item Tracer soigneusement la courbe repr\'{e}sentative $\mathcal{C}_{f}$ de
$f$ dans un rep\`{e}re orthonormal$\left(  O;\overrightarrow{i}%
;\overrightarrow{j}\right)  $ avec pour unit\'{e} $\left\Vert \overrightarrow
{i}\right\Vert =\left\Vert \overrightarrow{j}\right\Vert =1%
%TCIMACRO{\unit{cm}}%
%BeginExpansion
\operatorname{cm}%
%EndExpansion
.\label{Cf} $
\end{enumerate}

\item
\begin{enumerate}
\item Construire, \`{a} l'aide de la figure faite au \ref{Cf}), les huits
premiers termes de $\left(  u_{n}\right)  .$

\item Que peut-on conjecturer sur les variations de $\left(  u_{n}\right)
_{n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}}$?

\item On peut conjecturer que $u_{n}$ tend vers un certain nombre $l$ quand
$n$ tend vers $+\infty$.

\begin{enumerate}
\item d\'{e}terminer la valeur exacte de $l$ \emph{(On ne demande pas de
d\'{e}montrer que }$u_{n}$ \emph{converge vers }$l$\emph{)}.

\item V\'{e}rifier, \`{a} l'aide de votre calculatrice, que $u_{100}\simeq l$.
\end{enumerate}

\item
\begin{enumerate}
\item D\'{e}montrer, par r\'{e}currence, que:%
\[
\forall n\geq4,\quad u_{n}\in\left[  2;5\right]  .
\]


\item En d\'{e}duire que $\left(  u_{n}\right)  _{n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}}$ est born\'{e}e.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{exercise}


\end{document}