\section*{Part A}
Consider the function $f$ defined by :

$$f ( x ) = x - \ln ( 1 + x ) .$$

\begin{enumerate}
  \item Justify that the function $f$ is defined on the interval $] - 1 ; + \infty [$.
  \item We admit that the function $f$ is differentiable on $] - 1 ; + \infty [$.
\end{enumerate}

Determine the expression of its derivative function $f ^ { \prime } ( x )$.\\
3. a. Deduce the direction of variation of the function $f$ on the interval $] - 1 ; + \infty [$.\\
b. Deduce the sign of the function $f$ on the interval $] - 1 ; 0 [$.\\
4. a. Show that, for all $x$ belonging to the interval $] - 1 ; + \infty [$, we have :

$$f ( x ) = \ln \left( \frac { \mathrm { e } ^ { x } } { 1 + x } \right) .$$

b. Deduce the limit at $+ \infty$ of the function $f$.

\section*{Part B}
Consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 10$ and, for all natural integer $n$,

$$u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right) .$$

We admit that the sequence ( $u _ { n }$ ) is well defined.

\begin{enumerate}
  \item Give the value rounded to the nearest thousandth of $u _ { 1 }$.
  \item Using question 3. a. of Part A , prove by induction that, for all natural integer $n$, we have $u _ { n } \geqslant 0$.
  \item Prove that the sequence ( $u _ { n }$ ) is decreasing.
  \item Deduce from the previous questions that the sequence ( $u _ { n }$ ) converges.
  \item Determine the limit of the sequence $\left( u _ { n } \right)$.
\end{enumerate}