Let $f$ be the function defined and differentiable on the interval $[ 0 ; + \infty [$ such that:

$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } - x }$$

It is admitted that the function $f$ is positive on the interval $[ 0 ; + \infty [$.\\
We denote by $\mathscr { C }$ the representative curve of the function $f$ in an orthogonal coordinate system of the plane.\\
The curve $\mathscr { C }$ is represented in the appendix, to be returned with the answer sheet.

\section*{Part A}
Let the sequence $\left( I _ { n } \right)$ be defined for every natural integer $n$ by $I _ { n } = \int _ { 0 } ^ { n } f ( x ) \mathrm { d } x$.\\
We will not seek to calculate the exact value of $I _ { n }$ as a function of $n$.

\begin{enumerate}
  \item Show that the sequence ( $I _ { n }$ ) is increasing.
  \item It is admitted that for every real $x$ in the interval $\left[ 0 ; + \infty \left[ , \mathrm { e } ^ { x } - x \geqslant \frac { \mathrm { e } ^ { x } } { 2 } \right. \right.$.\\
a. Show that, for every natural integer $n , I _ { n } \leqslant \int _ { 0 } ^ { n } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x$.\\
b. Let $H$ be the function defined and differentiable on the interval $[ 0 ; + \infty [$ such that:
$$H ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }$$
Determine the derivative function $H ^ { \prime }$ of the function $H$.\\
c. Deduce that, for every natural integer $n , I _ { n } \leqslant 2$.
  \item Show that the sequence ( $I _ { n }$ ) is convergent. The value of its limit is not required.
\end{enumerate}

\section*{Part B}
Consider the following algorithm in which the variables are
\begin{itemize}
  \item $K$ and $i$ natural integers, $K$ being non-zero;
  \item $A , x$ and $h$ real numbers.
\end{itemize}

\begin{center}
\begin{tabular}{|l|l|}
\hline
Input: & Enter $K$ non-zero natural integer \\
\hline
Initialization & \begin{tabular}{l}
Assign to $A$ the value 0 \\
Assign to $x$ the value 0 \\
Assign to $h$ the value $\frac { 1 } { K }$ \\
\end{tabular} \\
\hline
Processing & \begin{tabular}{l}
For $i$ ranging from 1 to $K$ \\
Assign to $A$ the value $A + h \times f ( x )$ \\
Assign to $x$ the value $x + h$ \\
End For \\
\end{tabular} \\
\hline
Output & Display $A$ \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item Reproduce and complete the following table by running this algorithm for $K = 4$. The successive values of $A$ will be rounded to the nearest thousandth.

\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
$i$ & $A$ & $x$ \\
\hline
1 &  &  \\
\hline
2 &  &  \\
\hline
3 &  &  \\
\hline
4 &  &  \\
\hline
\end{tabular}
\end{center}

  \item By illustrating it on the appendix to be returned with the answer sheet, give a graphical interpretation of the result displayed by this algorithm for $K = 8$.
  \item What does the algorithm give when $K$ becomes large?
\end{enumerate}