Consider the function $f$ defined on $[0; +\infty[$ by
$$f(x) = \frac{\ln(x + 3)}{x + 3}$$

\begin{enumerate}
  \item Show that $f$ is differentiable on $[0; +\infty[$. Study the sign of its derivative function $f'$, its possible limit at $+\infty$, and draw up the table of its variations.
  \item We define the sequence $(u_n)_{n \geqslant 0}$ by its general term $u_n = \int_n^{n+1} f(x)\,\mathrm{d}x$.\\
  a. Justify that, if $n \leqslant x \leqslant n+1$, then $f(n+1) \leqslant f(x) \leqslant f(n)$.\\
  b. Show, without attempting to calculate $u_n$, that, for every natural integer $n$,
$$f(n+1) \leqslant u_n \leqslant f(n).$$
  c. Deduce that the sequence $(u_n)$ is convergent and determine its limit.
  \item Let $F$ be the function defined on $[0; +\infty[$ by
$$F(x) = [\ln(x+3)]^2.$$
  a. Justify the differentiability on $[0; +\infty[$ of the function $F$ and determine, for every positive real $x$, the number $F'(x)$.\\
  b. We set, for every natural integer $n$, $I_n = \int_0^n f(x)\,\mathrm{d}x$. Calculate $I_n$.
  \item We set, for every natural integer $n$, $S_n = u_0 + u_1 + \cdots + u_{n-1}$. Calculate $S_n$. Is the sequence $(S_n)$ convergent?
\end{enumerate}