\textbf{Exercise 5 — Candidates who have not chosen the specialisation option}

Let ( $v _ { n }$ ) be the sequence defined by

$$v _ { 1 } = \ln ( 2 ) \text { and, for all natural integer } n \text { non-zero, } v _ { n + 1 } = \ln \left( 2 - \mathrm { e } ^ { - v _ { n } } \right) .$$

We admit that this sequence is defined for all non-zero natural integer $n$.\\
We then define the sequence ( $S _ { n }$ ) for all non-zero natural integer $n$ by :

$$S _ { n } = \sum _ { k = 1 } ^ { n } v _ { k } = v _ { 1 } + v _ { 2 } + \cdots + v _ { n } .$$

The purpose of this exercise is to determine the limit of ( $S _ { n }$ ).

\section*{Part A - Conjectures using an algorithm}
\begin{enumerate}
  \item Copy and complete the following algorithm which calculates and displays the value of $S _ { n }$ for a value of $n$ chosen by the user :

\begin{center}
\begin{tabular}{|l|}
\hline
Variables : \\
$n , k$ integers \\
$S , v$ real numbers \\
Initialisation : \\
Input the value of $n$ \\
$v$ takes the value $\ldots$ \\
$S$ takes the value $\ldots$ \\
Processing: \\
For $k$ varying from \ldots to \ldots do \\
\ldots takes the value \ldots \\
\ldots takes the value \ldots \\
End For \\
Output : \\
Display $S$ \\
\hline
\end{tabular}
\end{center}

  \item Using this algorithm, we obtain some values of $S _ { n }$. The values rounded to the nearest tenth are given in the table below :

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
$n$ & 10 & 100 & 1000 & 10000 & 100000 & 1000000 \\
\hline
$S _ { n }$ & 2.4 & 4.6 & 6.9 & 9.2 & 11.5 & 13.8 \\
\hline
\end{tabular}
\end{center}

By explaining your approach, make a conjecture about the behaviour of the sequence $\left( S _ { n } \right)$.
\end{enumerate}

\section*{Part B - Study of an auxiliary sequence}
For all non-zero natural integer $n$, we define the sequence ( $u _ { n }$ ) by $u _ { n } = \mathrm { e } ^ { v _ { n } }$.

\begin{enumerate}
  \item Verify that $u _ { 1 } = 2$ and that, for all non-zero natural integer $n$, $u _ { n + 1 } = 2 - \frac { 1 } { u _ { n } }$.
  \item Calculate $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Results should be given in fractional form.
  \item Prove that, for all non-zero natural integer $n$, $u _ { n } = \frac { n + 1 } { n }$.
\end{enumerate}

\section*{Part C - Study of ( $S _ { n }$ )}
\begin{enumerate}
  \item For all non-zero natural integer $n$, express $v _ { n }$ as a function of $u _ { n }$, then $v _ { n }$ as a function of $n$.
  \item Verify that $S _ { 3 } = \ln ( 4 )$.
  \item For all non-zero natural integer $n$, express $S _ { n }$ as a function of $n$. Deduce the limit of the sequence $\left( S _ { n } \right)$.
\end{enumerate}