\textbf{Exercise 4 — Candidates who have NOT followed the specialization course}

We consider the sequence $(u_n)$ defined on $\mathbb{N}$ by:
$$u_0 = 2 \text{ and for all natural integer } n,\quad u_{n+1} = \frac{u_n + 2}{2u_n + 1}.$$
We admit that for all natural integer $n$, $u_n > 0$.

\begin{enumerate}
  \item a. Calculate $u_1, u_2, u_3, u_4$. An approximate value to $10^{-2}$ may be given.\\
b. Verify that if $n$ is one of the integers $0,1,2,3,4$ then $u_n - 1$ has the same sign as $(-1)^n$.\\
c. Establish that for all natural integer $n$, $u_{n+1} - 1 = \dfrac{-u_n + 1}{2u_n + 1}$.\\
d. Prove by induction that for all natural integer $n$, $u_n - 1$ has the same sign as $(-1)^n$.
  \item For all natural integer $n$, we set $v_n = \dfrac{u_n - 1}{u_n + 1}$.\\
a. Establish that for all natural integer $n$, $v_{n+1} = \dfrac{-u_n + 1}{3u_n + 3}$.\\
b. Prove that the sequence $(v_n)$ is a geometric sequence with ratio $-\dfrac{1}{3}$. Deduce the expression of $v_n$ as a function of $n$.\\
c. We admit that for all natural integer $n$, $u_n = \dfrac{1 + v_n}{1 - v_n}$. Express $u_n$ as a function of $n$ and determine the limit of the sequence $(u_n)$.
\end{enumerate}