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\def\examtitle{Devoir 02 - S01}
\def\schoolname{\textbf{Lycée :} Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{: (6.5 pts)}{
On considère la suite réelle \((u_n)\) définie par :
~$
u_0 = 5 \quad \text{et} \quad (\forall n \in \mathbb{N}) \quad u_{n+1} = \frac{5u_n - 4}{u_n}
$~
\begin{enumerate}[label=\arabic*)]
    \item Calculer \(u_1\) puis montrer que : \((\forall n \in \mathbb{N}) : u_n > 4\). \hfill (1 point)

    \item
    \begin{enumerate}[label=(\alph*)]
        \item Vérifier que : \(u_{n+1} - u_n = \frac{(u_n - 1)(4 - u_n)}{u_n}\) pour tout \(n \in \mathbb{N}\). \hfill (0,75 point)
        \item Montrer que la suite \((u_n)\) est décroissante. En déduire que : \((\forall n \in \mathbb{N}) : u_n \leq 5\). \hfill (0,75 point)
    \end{enumerate}

    \item Soit \((v_n)_{n \in \mathbb{N}}\) la suite réelle définie par : \(v_n = \frac{u_n - 4}{u_n - 1}\).
    \begin{enumerate}[label=(\alph*)]
        \item Montrer que \((v_n)\) est une suite géométrique de raison \(q = \frac{1}{4}\). \hfill (1 point)
        \item Écrire \(v_n\) en fonction de \(n\). \hfill (0,5 point)
        \item Montrer que : \((\forall n \in \mathbb{N}) : u_n = \frac{1 - 4^{n+2}}{1 - 4^{n+1}}\), puis calculer \(\lim\limits_{n \to +\infty} u_n\). \hfill (1 point)
    \end{enumerate}

    \item On pose pour tout \(n \in \mathbb{N}^* : S_n = v_0 + v_1 + \cdots + v_n\) et \(T_n = \cos(\pi S_n)\).
    \begin{enumerate}[label=(\alph*)]
        \item Montrer que : \((\forall n \in \mathbb{N}) : S_n = \frac{1}{3} \left(1 - \left(\frac{1}{4}\right)^{n+1}\right)\). \hfill (1 point)
        \item En déduire la limite de la suite \((T_n)\). \hfill (0,5 point)
    \end{enumerate}
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{: (13 pts)}{
On considère la fonction numérique \( f \) définie par : \( f(x) = \dfrac{x+1}{2\sqrt{x}} \) et \( (C_f) \) sa courbe représentative dans un repère orthonormé \( (O; \vec{i}, \vec{j}) \) telles que : \( \|\vec{i}\| = \|\vec{j}\| = 1\text{cm} \)
\begin{enumerate}[label=\arabic*)]
    \item Vérifier que \( D_f = ]0; +\infty[ \). \dotfill (0,5 pt)
    \item Montrer que : \( \displaystyle\lim_{x \to 0^+} f(x) = +\infty \), puis interpréter le résultat géométriquement. \dotfill (1 pt)
    \item Calculer : \( \displaystyle\lim_{x \to +\infty} f(x) \) et \( \displaystyle\lim_{x \to +\infty} \dfrac{f(x)}{x} \), puis en déduire la nature de la branche infinie de \( (C_f) \) au voisinage de \( +\infty \). \dotfill (1 pt)
    \item
      \begin{enumerate}
          \item
      Montrer que : \( (\forall x \in D_f) : f(x) - x = \dfrac{(1-\sqrt{x})(2x+\sqrt{x}+1)}{2\sqrt{x}} \). \dotfill (1 pt)
    \item En déduire la position relative de \( (C_f) \) et la droite \( (\Delta) : y = x \). \dotfill (1 pt)
      \end{enumerate}
    \item
      \begin{enumerate}
    \item Montrer que : \( (\forall x \in ]0; +\infty[) : f'(x) = \dfrac{x-1}{4\sqrt{x^3}} \). \dotfill (1 pt)
    \item En déduire que \( f \) est strictement croissante sur \( [1; +\infty[ \) et décroissante sur \( ]0; 1] \). \dotfill (1 pt)
      \end{enumerate}
    \item
      \begin{enumerate}
    \item Montrer que : \( (\forall x \in ]0; +\infty[) : f''(x) = \dfrac{3-x}{8\sqrt{x^5}} \). \dotfill (1 pt)
    \item Déduire la concavité de \( (C_f) \) et déterminer son point d'inflexion s'il existe. \dotfill (1 pt)
      \end{enumerate}
    \item
      \begin{enumerate}
        \item Déterminer les primitives de \( f \) sur \( ]0; +\infty[ \) (\textbf{remarquez que} : \( f(x) = \dfrac{\sqrt{x}}{2} + \dfrac{1}{2\sqrt{x}} \)). \dotfill (1 pt)
    \item En déduire la fonction primitive \( F \) de \( f \) vérifiant : \( F(1) = 0 \). \dotfill (1 pt)
      \end{enumerate}
    \item Construire dans le même repère la droite \( (\Delta) \) et la courbe \( (C_f) \). \dotfill (1 pt)
    \item On considère la suite \( (u_n) \) définie par : \( u_0 = 4 \), \( u_{n+1} = f(u_n) \) avec \( n \in \mathbb{N} \).
    \begin{enumerate}[label=\alph*)]
        \item Montrer que : \( (\forall n \in \mathbb{N}) : u_n \geq 1 \). \dotfill (1 pt)
        \item Montrer que \( (u_n) \) est décroissante. \dotfill (0,5 pt)
        \item En déduire que la suite \( (u_n) \) est convergente, puis calculer \( \displaystyle\lim_{n \to +\infty} u_n \). \dotfill (1 pt)
    \end{enumerate}
\end{enumerate}
}








\end{document}