We consider the sequence $\left( u _ { n } \right)$ defined, for all non-zero natural integers $n$, by:

$$u _ { n } = \frac { n ( n + 2 ) } { ( n + 1 ) ^ { 2 } }$$

The sequence $( v _ { n } )$ is defined by:
$v _ { 1 } = u _ { 1 }$, $v _ { 2 } = u _ { 1 } \times u _ { 2 }$ and for all natural integers $n \geqslant 3$, $v _ { n } = u _ { 1 } \times u _ { 2 } \times \ldots \times u _ { n } = v _ { n - 1 } \times u _ { n }$.

\begin{enumerate}
  \item Verify that we have $v _ { 2 } = \frac { 2 } { 3 }$ then calculate $v _ { 3 }$.
  \item We consider the incomplete algorithm below. Copy and complete this algorithm on your paper so that, after its execution, the variable $V$ contains the value $v _ { n }$ where $n$ is a non-zero natural integer defined by the user. No justification is required.

\begin{center}
\begin{tabular}{ | l l | }
\hline
 & Algorithm \\
\hline
1. & $V \leftarrow 1$ \\
2. & For $i$ varying from 1 to $n$ \\
3. & $U \leftarrow \frac { \ldots ( \ldots + 2 ) } { ( \ldots + 1 ) ^ { 2 } }$ \\
4. & $V \leftarrow \ldots$ \\
5. & End For \\
\hline
\end{tabular}
\end{center}

  \item a. Show that, for all non-zero natural integers $n$, $u _ { n } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$.\\
b. Show that, for all non-zero natural integers $n$, $0 < u _ { n } < 1$.
  \item a. Show that the sequence $( v _ { n } )$ is decreasing.\\
b. Justify that the sequence $( v _ { n } )$ is convergent (we do not ask to calculate its limit).
  \item a. Verify that, for all non-zero natural integers $n$, $v _ { n + 1 } = v _ { n } \times \frac { ( n + 1 ) ( n + 3 ) } { ( n + 2 ) ^ { 2 } }$.\\
b. Show by induction that, for all non-zero natural integers $n$, $v _ { n } = \frac { n + 2 } { 2 ( n + 1 ) }$.\\
c. Determine the limit of the sequence $\left( v _ { n } \right)$.
  \item We consider the sequence $w _ { n }$ defined by\\
$w _ { 1 } = \ln \left( u _ { 1 } \right)$, $w _ { 2 } = \ln \left( u _ { 1 } \right) + \ln \left( u _ { 2 } \right)$ and, for all natural integers $n \geqslant 3$, by
$$w _ { n } = \sum _ { k = 1 } ^ { n } \ln \left( u _ { k } \right) = \ln \left( u _ { 1 } \right) + \ln \left( u _ { 2 } \right) + \ldots + \ln \left( u _ { n } \right)$$
Show that $w _ { 7 } = 2 w _ { 1 }$.
\end{enumerate}