\textbf{Exercise 2 Sequences}

Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 4$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{5} u_{n}^{2}$.

\begin{enumerate}
  \item a. Calculate $u_{1}$ and $u_{2}$.\\
b. Copy and complete the function below written in Python language. This function is named suite\_u and takes as parameter the natural integer $p$. It returns the value of the term of rank $p$ of the sequence $(u_{n})$.
\begin{verbatim}
def suite_u(p) :
    u= ...
    for i in range(1,...) :
        u =...
    return u
\end{verbatim}
  \item a. Prove by induction that for every natural integer $n$, $0 < u_{n} \leqslant 4$.\\
b. Prove that the sequence $(u_{n})$ is decreasing.\\
c. Deduce from this that the sequence $(u_{n})$ is convergent.
  \item a. Justify that the limit $\ell$ of the sequence $(u_{n})$ satisfies the equality $\ell = \frac{1}{5} \ell^{2}$.\\
b. Deduce from this the value of $\ell$.
  \item For every natural integer $n$, we set $v_{n} = \ln\left(u_{n}\right)$ and $w_{n} = v_{n} - \ln(5)$.\\
a. Show that, for every natural integer $n$, $v_{n+1} = 2v_{n} - \ln(5)$.\\
b. Show that the sequence $(w_{n})$ is geometric with common ratio 2.\\
c. For every natural integer $n$, give the expression of $w_{n}$ as a function of $n$ and show that $v_{n} = \ln\left(\frac{4}{5}\right) \times 2^{n} + \ln(5)$.
  \item Calculate $\lim_{n \rightarrow +\infty} v_{n}$ and find again $\lim_{n \rightarrow +\infty} u_{n}$.
\end{enumerate}