We equip the plane with an orthonormal coordinate system. For every natural integer $n$, we consider the function $f_n$ defined on $[0; +\infty[$ by:
$$f_0(x) = \mathrm{e}^{-x} \text{ and, for } n \geqslant 1,\ f_n(x) = x^n \mathrm{e}^{-x}.$$
For every natural integer $n$, we denote $\mathscr{C}_n$ the representative curve of the function $f_n$.

Parts A and B are independent.

\textbf{Part A: Study of the functions $f_n$ for $n \geqslant 1$}

We consider a natural integer $n \geqslant 1$.
\begin{enumerate}
  \item a. We admit that the function $f_n$ is differentiable on $[0; +\infty[$. Show that for all $x \geqslant 0$,
$$f_n'(x) = (n - x)x^{n-1}\mathrm{e}^{-x}.$$
b. Justify all elements of the table below:
\begin{center}
\begin{tabular}{ | c | l l c l l | }
\hline
$x$ & 0 &  & $n$ &  & $+\infty$ \\
\hline
$f_n'(x)$ &  & + & 0 & - &  \\
\hline
 &  &  & $\left(\frac{n}{\mathrm{e}}\right)^n$ &  &  \\
$f_n$ &  &  &  &  &  \\
 & 0 &  &  &  & 0 \\
\hline
\end{tabular}
\end{center}
  \item Justify by calculation that the point $\mathrm{A}\left(1; \mathrm{e}^{-1}\right)$ belongs to the curve $\mathscr{C}_n$.
\end{enumerate}

\textbf{Part B: Study of the integrals $\int_0^1 f_n(x)\,\mathrm{d}x$ for $n \geqslant 0$}

In this part, we study the functions $f_n$ on $[0; 1]$ and we consider the sequence $(I_n)$ defined for every natural integer $n$ by:
$$I_n = \int_0^1 f_n(x)\,\mathrm{d}x = \int_0^1 x^n \mathrm{e}^{-x}\,\mathrm{d}x.$$

\begin{enumerate}
  \item On the graph in APPENDIX, the curves $\mathscr{C}_0, \mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_{10}$ and $\mathscr{C}_{100}$ are represented.\\
a. Give a graphical interpretation of $I_n$.\\
b. By reading this graph, what conjecture can be made about the limit of the sequence $(I_n)$?
  \item Calculate $I_0$.
  \item a. Let $n$ be a natural integer. Prove that for all $x \in [0; 1]$,
$$0 \leqslant x^{n+1} \leqslant x^n.$$
b. Deduce that for every natural integer $n$, we have:
$$0 \leqslant I_{n+1} \leqslant I_n.$$
  \item Prove that the sequence $(I_n)$ is convergent, towards a limit greater than or equal to zero that we will denote $\ell$.
  \item Using integration by parts, prove that for every natural integer $n$ we have:
$$I_{n+1} = (n+1)I_n - \frac{1}{\mathrm{e}}.$$
  \item a. Prove that if $\ell > 0$, the equality from question 5 leads to a contradiction.\\
b. Prove that $\ell = 0$. You may use question 6.a.
  \item The script of the \texttt{mystere} function is given below, written in Python language. The constant \texttt{e} has been imported.
\begin{verbatim}
def mystere(n):
    I = 1 - 1/e
    L = [I]
    for i in range(n):
        I = (i + 1)*I - 1/e
        L.append(I)
    return L
\end{verbatim}
What does \texttt{mystere(100)} return in the context of the exercise?
\end{enumerate}