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%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 13165701020638--
%%--Corrigé du document N° 1316569596882--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{claim}
$\vspace*{-0.35cm}$

\begin{enumerate}
\item
\begin{enumerate}
\item La\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{1 pt}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{1 pt}}%
%EndExpansion
} fonction $f:x\mapsto\dfrac{2x+1}{x-1}$ est d\'{e}finie et d\'{e}rivable sur
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
/\left\{  1\right\}  $ et:%
\[
f^{\prime}(x)=\dfrac{2(x-1)-(2x+1)}{(x-1)^{2}}=\dfrac{-3}{(x-1)^{2}}%
<0\quad\forall x\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
/\left\{  1\right\}  .
\]


D'o\`{u} le tableau de variation de $f:$%
\[%
\begin{tabular}
[c]{c|ccc||lcc}%
$x$ & $-\infty$ &  & \multicolumn{2}{c}{$1$} &  & $+\infty$\\\hline
&  &  &  &  &  & \\
$f^{\prime}(x)$ &  & $-$ &  &  & $-$ & \\
&  &  &  &  &  & \\\hline
&  &  &  &  &  & \\
& $2$ &  &  & $+\infty$ &  & \\
$f$ &  &
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&  &  &
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& \\
&  &  & $-\infty$ &  &  & $2$%
\end{tabular}
\]


\item Voir\marginpar{%
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%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{1 pt}}%
%EndExpansion
} figure ci-apr\`{e}s.
\end{enumerate}

\item
\begin{enumerate}
\item Voir\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{2 pts}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{2 pts}}%
%EndExpansion
} figure ci-apr\`{e}s$.$

\item On\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{1 pt}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{1 pt}}%
%EndExpansion
} peut conjecturer que $(u_{n})$ n'est pas monotone.

\item
\begin{enumerate}
\item On\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{1 pt}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{1 pt}}%
%EndExpansion
} peut conjecturer que $\underset{n\rightarrow+\infty}{\lim}u_{n}=l\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$ et comme $\underset{n\rightarrow+\infty}{\lim}u_{n}=\underset
{n\rightarrow+\infty}{\lim}u_{n+1},$ on en d\'{e}duit que: $l=\dfrac
{2l+1}{l-1}.$

Or%
\begin{align*}
l  &  =\dfrac{2l+1}{l-1}\\
&  \Leftrightarrow l(l-1)=2l+1\\
&  \Leftrightarrow l^{2}-3l-1=0\\
\overset{\Delta=13}{\Leftrightarrow}l  &  =\dfrac{3+\sqrt{13}}{2}\quad ou\quad
l=\dfrac{3-\sqrt{13}}{2}\quad\text{{\small (qui est n\'{e}gatif)}}.
\end{align*}
%

\[
\fbox{Donc, si $u_{n}$ converge vers $l>0$ alors $l=\dfrac{\sqrt{13}+3}{2}$}%
\]


\item $\quad$\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{1 pt}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{1 pt}}%
%EndExpansion
}%
\[
\fbox{La calculatrice donne $u_{100}\simeq3.3027756$ et $\dfrac{\sqrt{13}%
+3}{2}\simeq3.3027756;$ ce qui conforte notre conjecture.}%
\]


\item D\'{e}montrons\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{2 pts}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{2 pts}}%
%EndExpansion
} par r\'{e}currence que la propri\'{e}t\'{e} $P_{n}:$"$u_{n}\in\left[
2;5\right]  $" est vraie pour tout $n\geq4:$

-\textbf{\ Initialisation:}

$\quad$On a
\begin{align*}
u_{0}  &  =1/2;\\
u_{1}  &  =\dfrac{2u_{0}+1}{u_{0}-1}=\dfrac{2\dfrac{1}{2}+1}{\dfrac{1}{2}%
-1}=-7;\\
u_{2}  &  =\dfrac{2u_{1}+1}{u_{1}-1}=\dfrac{2(-7)+1}{-7-1}=\dfrac{13}{8};\\
u_{3}  &  =\dfrac{2u_{2}+1}{u_{2}-1}=\dfrac{2\dfrac{13}{8}+1}{\dfrac{13}{8}%
-1}=\dfrac{37}{5};\\
u_{4}  &  =\dfrac{2u_{3}+1}{u_{3}-1}=\dfrac{2\dfrac{37}{5}+1}{\dfrac{37}{5}%
-1}=\dfrac{13}{8}=1.625
\end{align*}


Donc $u_{4}\in\lbrack2;5]\quad$ie: $P_{4}$ est vraie.

-\textbf{\ H\'{e}r\'{e}dit\'{e}:}

$\quad$Supposons que $P_{n}$ \`{a} un certain rang $n\geq4$ ie $u_{n}%
\in\lbrack2;5]$ pour un certain $n\geq4;$ on a alors:%
\begin{align*}
2  &  \leq u_{n}\leq5\\
&  \Longrightarrow f(2)\geq f(u_{n})\geq f(5)\quad\text{car }f\text{ est
d\'{e}croissante sur }[2;5]\\
&  \Leftrightarrow\dfrac{2\times2+1}{2-1}\geq u_{n+1}\geq\dfrac{2\times
5+1}{5-1}\\
&  \Leftrightarrow5\geq u_{n+1}\geq\dfrac{11}{4}=2.75.
\end{align*}


On a donc bien: $2\leq u_{n+1}\leq5\quad$ie $P_{n+1}$ est vraie.

-\textbf{\ Conclusion:}

$\quad$On a d\'{e}montr\'{e} que $P_{4}$ est vraie et que si $P_{n}$ est vraie
\`{a} un certain rang $n\geq4$ alors $P_{n+1}$ est aussi vraie, donc:%
\[
\fbox{$P_{n}$ est vraie pour tout $n\geq4\quad$ie: $\forall n\geq4,u_{n}%
\in\lbrack2;5].$}%
\]


\item $(u_{n})$ est born\'{e}e pour $n\in\{0;1;2;3\}$ ($-7\leq u_{n}\leq
\tfrac{37}{5}$ pour $0\leq n\leq3$) et pour $n\geq4$ (d'apr\`{e}s la question
pr\'{e}c\'{e}dente) \fbox{donc $\left(  u_{n}\right)  $ est born\'{e}e sur ${%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
.}$}
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{claim}

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