\textbf{Exercise 2 — Candidates WHO HAVE NOT FOLLOWED the mathematics specialization course}

We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$,
$$u _ { n + 1 } = \sqrt { 2 u _ { n } } .$$

\begin{enumerate}
  \item We consider the following algorithm:
\end{enumerate}

\begin{center}
\begin{tabular}{ | l l | }
\hline
Variables: & $n$ is a natural integer \\
 & $u$ is a positive real number \\
Initialization: & Request the value of $n$ \\
 & Assign to $u$ the value 1 \\
Processing: & For $i$ varying from 1 to $n :$ \\
 & $\mid$ Assign to $u$ the value $\sqrt { 2 u }$ \\
 & End of For \\
Output : & Display $u$ \\
\hline
\end{tabular}
\end{center}

a. Give an approximate value to $10 ^ { - 4 }$ of the result displayed by this algorithm when $n = 3$ is chosen.\\
b. What does this algorithm allow us to calculate?\\
c. The table below gives approximate values obtained using this algorithm for certain values of $n$.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$n$ & 1 & 5 & 10 & 15 & 20 \\
\hline
Displayed value & 1,4142 & 1,9571 & 1,9986 & 1,9999 & 1,9999 \\
\hline
\end{tabular}
\end{center}

What conjectures can be made concerning the sequence $\left( u _ { n } \right)$ ?\\
2. a. Prove that, for every natural integer $n , 0 < u _ { n } \leqslant 2$.\\
b. Determine the direction of variation of the sequence $( u _ { n } )$.\\
c. Prove that the sequence $( u _ { n } )$ is convergent. We do not ask for the value of its limit.\\
3. We consider the sequence $( v _ { n } )$ defined, for every natural integer $n$, by $v _ { n } = \ln u _ { n } - \ln 2$.\\
a. Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio $\frac { 1 } { 2 }$ and first term $v _ { 0 } = - \ln 2$.\\
b. Determine, for every natural integer $n$, the expression of $v _ { n }$ as a function of $n$, then of $u _ { n }$ as a function of $n$.\\
c. Determine the limit of the sequence $\left( u _ { n } \right)$.\\
d. Copy the algorithm below and complete it with the instructions for processing and output, so as to display as output the smallest value of $n$ such that $u _ { n } > 1,999$.

\begin{center}
\begin{tabular}{ | l l | }
\hline
Variables: & $n$ is a natural integer \\
 & $u$ is a real number \\
Initialization : & Assign to $n$ the value 0 \\
 & Assign to $u$ the value 1 \\
Processing: & \\
Output : & \\
\hline
\end{tabular}
\end{center}