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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1315038195168--
%%--Utilisation d\'une suite auxiliaire pour expliciter une suite--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}[D'après un sujet de BAC]
La suite $\left(  u_{n}\right)  $ est définie par:%
\[
u_{0}=0\quad\text{et\quad}\forall n\in%
\mathbb{N}
,\text{ }u_{n+1}=\frac{2u_{n}+1}{u_{n}+2}.
\]

\begin{enumerate}
\item Démontrer, par récurrence, que: $\forall n\in \mathbb{N},$ $0\leq
  u_n <1.$
\item Démontrer que la suite $\left(u_{n}\right)  $ est monotone.

\item La suite $\left(  v_{n}\right)  $ est définie pour tout entier $n$ par
$v_{n}=\dfrac{u_{n}-1}{u_{n}+1}.$

\begin{enumerate}
\item Démontrer que $\left(  v_{n}\right)  $ est une suite géométrique;
préciser sa raison et son premier terme.

\item Exprimer $v_{n}$, puis $u_{n}$, en fonction de $n$ et déterminer
$\underset{n\rightarrow+\infty}{\lim}u_{n}.$
\end{enumerate}

\item Trouver un entier $N$ tel que, pour tout $n>N$ on ait $u_{n}>0,99$.
\end{enumerate}
\end{exercise}


\end{document}