\textbf{Exercise B} (Main topics: Sequences, function study, Logarithm function)

Let the function $f$ defined on the interval $]1; +\infty[$ by
$$f(x) = x - \ln(x-1).$$
Consider the sequence $(u_n)$ with initial term $u_0 = 10$ and such that $u_{n+1} = f(u_n)$ for every natural integer $n$.

\section*{Part I:}
The spreadsheet below was used to obtain approximate values of the first terms of the sequence $(u_n)$.

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
 & A & B \\
\hline
1 & $n$ & $u_n$ \\
\hline
2 & 0 & 10 \\
\hline
3 & 1 & 7.80277542 \\
\hline
4 & 2 & 5.88544474 \\
\hline
5 & 3 & 4.29918442 \\
\hline
6 & 4 & 3.10550913 \\
\hline
7 & 5 & 2.36095182 \\
\hline
8 & 6 & 2.0527675 \\
\hline
9 & 7 & 2.00134509 \\
\hline
10 & 8 & 2.0000009 \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item What formula was entered in cell B3 to allow the calculation of approximate values of $(u_n)$ by copying downward?
  \item Using these values, conjecture the direction of variation and the limit of the sequence $(u_n)$.
\end{enumerate}

\section*{Part II:}
We recall that the function $f$ is defined on the interval $]1; +\infty[$ by
$$f(x) = x - \ln(x-1).$$

\begin{enumerate}
  \item Calculate $\lim_{x \rightarrow 1} f(x)$. We will admit that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.
  \item a. Let $f^{\prime}$ be the derivative function of $f$. Show that for all $x \in ]1; +\infty[$, $f^{\prime}(x) = \frac{x-2}{x-1}$.\\
b. Deduce the table of variations of $f$ on the interval $]1; +\infty[$, completed by the limits.\\
c. Justify that for all $x \geqslant 2$, $f(x) \geqslant 2$.
\end{enumerate}

\section*{Part III:}
\begin{enumerate}
  \item Using the results of Part II, prove by induction that $u_n \geqslant 2$ for every natural integer $n$.
  \item Show that the sequence $(u_n)$ is decreasing.
  \item Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$.
  \item We admit that $\ell$ satisfies $f(\ell) = \ell$. Give the value of $\ell$.
\end{enumerate}