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.
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 $n$.
- Show that the sequence ( $I _ { n }$ ) is increasing.
- 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 $n , I _ { n } \leqslant 2$.
- Show that the sequence ( $I _ { n }$ ) is convergent. The value of its limit is not required.
Part B
Consider the following algorithm in which the variables are
- $K$ and $i$ natural integers, $K$ being non-zero;
- $A , x$ and $h$ real numbers.
| Input: | Enter $K$ non-zero natural integer |
| Initialization | \begin{tabular}{l} Assign to $A$ the value 0 |
| Assign to $x$ the value 0 |
| Assign to $h$ the value $\frac { 1 } { K }$ |
\hline Processing &
| 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 |
\hline Output & Display $A$ \hline \end{tabular}
- 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.
- 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 $K = 8$.
- What does the algorithm give when $K$ becomes large?