The sequence ( $u _ { n }$ ) is defined by: $$u _ { 0 } = 0 \quad \text { and, for all natural integer } n , u _ { n + 1 } = \frac { 1 } { 2 - u _ { n } } .$$
a. Using the calculation of the first terms of the sequence ( $u _ { n }$ ), conjecture the explicit form of $u _ { n }$ as a function of $n$. Prove this conjecture. b. Deduce the value of the limit $\ell$ of the sequence $\left( u _ { n } \right)$.
Complete, in appendix 2, the algorithm to determine the value of the smallest integer $n$ such that $\left| u _ { n + 1 } - u _ { n } \right| \leqslant 10 ^ { - 3 }$.
The sequence ( $u _ { n }$ ) is defined by:
$$u _ { 0 } = 0 \quad \text { and, for all natural integer } n , u _ { n + 1 } = \frac { 1 } { 2 - u _ { n } } .$$
\begin{enumerate}
\item a. Using the calculation of the first terms of the sequence ( $u _ { n }$ ), conjecture the explicit form of $u _ { n }$ as a function of $n$. Prove this conjecture.\\
b. Deduce the value of the limit $\ell$ of the sequence $\left( u _ { n } \right)$.
\item Complete, in appendix 2, the algorithm to determine the value of the smallest integer $n$ such that $\left| u _ { n + 1 } - u _ { n } \right| \leqslant 10 ^ { - 3 }$.
\end{enumerate}