\section*{Part A}
In the plane with an orthonormal coordinate system, we denote by $\mathscr { C } _ { 1 }$ the curve representing the function $f _ { 1 }$ defined on $\mathbb { R }$ by:
$$f _ { 1 } ( x ) = x + \mathrm { e } ^ { - x } .$$

\begin{enumerate}
  \item Justify that $\mathscr { C } _ { 1 }$ passes through point A with coordinates $( 0 ; 1 )$.
  \item Determine the variation table of the function $f _ { 1 }$. Specify the limits of $f _ { 1 }$ at $+ \infty$ and at $- \infty$.
\end{enumerate}

\section*{Part B}
The purpose of this part is to study the sequence $\left( I _ { n } \right)$ defined on $\mathbb { N }$ by:
$$I _ { n } = \int _ { 0 } ^ { 1 } \left( x + \mathrm { e } ^ { - n x } \right) \mathrm { d } x .$$

\begin{enumerate}
  \item In the plane with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ), for every natural integer $n$, we denote by $\mathscr { C } _ { n }$ the curve representing the function $f _ { n }$ defined on $\mathbb { R }$ by
$$f _ { n } ( x ) = x + \mathrm { e } ^ { - n x } .$$
a. Give a geometric interpretation of the integral $I _ { n }$.\\
b. Using this interpretation, formulate a conjecture about the direction of variation of the sequence ( $I _ { n }$ ) and its possible limit. Specify the elements on which you base your conjecture.\\
  \item Prove that for every natural integer $n$ greater than or equal to 1,
$$I _ { n + 1 } - I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - ( n + 1 ) x } \left( 1 - \mathrm { e } ^ { x } \right) \mathrm { d } x$$
Deduce the sign of $I _ { n + 1 } - I _ { n }$ and then prove that the sequence ( $I _ { n }$ ) is convergent.\\
  \item Determine the expression of $I _ { n }$ as a function of $n$ and determine the limit of the sequence $\left( I _ { n } \right)$.
\end{enumerate}