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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
\textbf{Nom : }$\dots\dots\dots\dots\dots$ \hfill \textbf{Pr\'enom : }$\dots\dots\dots\dots\dots$  \hfill Classe :$\dots\dots$\\
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\begin{minipage}{17cm}
\begin{center}
\vspace{1mm}
\textsc{\textbf{ Interrogation n$^{\circ}16$ }}
\end{center}
\vspace{-3mm}
\end{minipage}
}
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\begin{center}
\textit{\textbf{On prendra soin de coller le sujet sur la copie. La note tiendra compte de la qualité de la rédaction et des justifications.}}
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\medskip
 
 
 \Exoi{\hfill (10 points)\\
 Soit $\left(u_{n}\right)$ la suite définie par 	$$\left\{\begin{array}{l c l}
u_{0}& =& 1\\
u_{n+1}&=&u_{n}- \ln \left(u_{n}^2 + 1\right)~ \text{pour tout entier naturel}~ n.
\end{array}\right.$$
\partieBac{}
\begin{enumerate}
\item Construire, en utilisant la droite d'équation $y=x$ et la courbe d'équation $y=x-\ln(x^2+1)$,   les termes $u_0$, $u_1$, $u_2$ et $u_3$ sur l'axe des abscisses.
\item Que peut-on conjecturer pour la suite $(u_n)_{n\in \N}$ ?
\end{enumerate}
\partieBac{}
Soit $f$ la fonction définie sur $\R$ par 
\[f(x)= x - \ln \left(x^2 + 1\right).\] 
\begin{enumerate}
\item Résoudre dans $\R$ l'équation $f(x) = x$. 
\item Démontrer que pour $x\in [0;1]$ on a :
$$ f'(x)=\dfrac{(x-1)^2}{x^2+1}$$
Dresser le tableau de variation complet de $f$ sur $[0;1]$ et déduire que si $x \in [0~;~1]$ alors $f(x) \in [0~;~1]$.
\end{enumerate}
\partieBac{}
\begin{enumerate}
\item Démontrer par récurrence que, pour tout entier $n \geqslant  0,~ u_{n} \in [0~;~1]$. 
\item Étudier le sens de variation de la suite $\left(u_{n}\right)$. 
\item Démontrer que la suite $\left(u_{n}\right)$ est convergente. Déterminer sa limite. 
\end{enumerate} 
}

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%\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
%\textbf{Nom : }$\dots\dots\dots\dots\dots$ \hfill \textbf{Pr\'enom : }$\dots\dots\dots\dots\dots$  \hfill Classe :$\dots\dots$\\
%\psframebox[linestyle=solid,shadow=true,framearc=0.5,fillcolor=vert!30]{
%\begin{minipage}{17cm}
%\begin{center}
%\vspace{1mm}
%\textsc{\textbf{ Interrogation n$^{\circ}16$ }}
%\end{center}
%\vspace{-3mm}
%\end{minipage}
%}
%\setcounter{Exi}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	 Interro
%\begin{center}
%\textit{\textbf{On prendra soin de coller le sujet sur la copie. La note tiendra compte de la qualité de la rédaction et des justifications.}}
%\end{center}
%

\end{document}
