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\def\examtitle{Série: Limite d'une suite numérique}
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% ----------------------------------------------------------------------
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% ----------------------------------------------------------------------

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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
Soit \((U_n)\) la suite numérique définie par :
~$
\begin{cases}
U_{n+1} = \dfrac{3U_n - 8}{2U_n - 5}\quad , & (\forall n \in \mathbb{N}) \\
U_0 = 4
\end{cases}
$~

\begin{enumerate}[label=\arabic*.]
    \item Soit \(n \in \mathbb{N}\) on pose :
    ~$
    V_n = \dfrac{U_n - 3}{U_n - 2}
    $~
    \begin{enumerate}[label=\alph*)]
        \item Vérifier que :
        ~$
        (\forall n \in \mathbb{N}) \quad  V_n + 2 = \dfrac{3U_n - 7}{U_n - 2}
        $~
        \item Montrer que la suite \((V_n)\) est arithmétique de raison \(r = 2\).
        \item Calculer \(V_0\), puis déterminer \(V_n\) en fonction de \(n\).
    \end{enumerate}
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) \quad  U_n = \dfrac{8n - 4}{4n - 1}, \text{ puis calculer } \lim U_n.
    $~
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
Soit \((U_n)\) la suite numérique définie par :
~$
U_0 = 3 \quad \text{et} \quad U_{n+1} = \dfrac{8(U_n - 1)}{U_n + 2}; \quad (\forall n \in \mathbb{N})
$~

\begin{enumerate}[label=\arabic*.]
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N})\quad  : 2 < U_n < 4.
    $~
    \item Déterminer la monotonie de la suite \((U_n)\) et vérifier qu’elle est convergente.
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) \quad  4 - U_{n+1} \leq \dfrac{4}{5}(4 - U_n).
    $~
    \item En déduire que :
    ~$
    (\forall n \in \mathbb{N}) \quad  4 - U_n \leq \left( \dfrac{4}{5} \right)^n, \text{ puis calculer } \lim U_n.
    $~
    \item Soit \(n \in \mathbb{N}\) on pose :
    ~$
    V_n = \dfrac{U_n - 4}{U_n - 2}.
    $~
    \item Montrer que \((V_n)\) est géométrique.
    \item Redéterminer \(\lim U_n\).
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{}{
Soit \(f\) la fonction numérique définie sur \(I = [1, 2]\) par :
~$
f(x) = \dfrac{3x + 2}{2x + 3}.
$~

\begin{enumerate}[label=\arabic*.]
    \item Étudier les variations de la fonction \(f\) sur l’intervalle \(I\).
    \item En déduire l’image de l’intervalle \(I\) par la fonction \(f\).
    \item Montrer que :
    ~$
    (\forall x \in I)\quad  : f(x) \leq x
    $~
\end{enumerate}

On considère \((U_n)\) la suite numérique définie par :
~$
U_0 = 2 \quad \text{et} \quad (\forall n \in \mathbb{N}) \quad  U_{n+1} = f(U_n)
$~

\begin{enumerate}[resume]
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) \quad : 1 \leq U_n \leq 2.
    $~
    \item Montrer que la suite \((U_n)\) est décroissante.
    \item En déduire que la suite \((U_n)\) est convergente puis déterminer \(\lim U_n\).
\end{enumerate}
}






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\end{document}