We consider the sequence $\left( I _ { n } \right)$ defined by $I _ { 0 } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 - x } \mathrm {~d} x$ and for every non-zero natural number $n$

$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { x ^ { n } } { 1 - x } \mathrm {~d} x$$

\begin{enumerate}
  \item Show that $I _ { 0 } = \ln ( 2 )$.
  \item a. Calculate $I _ { 0 } - I _ { 1 }$.\\
b. Deduce $I _ { 1 }$.
  \item a. Show that, for every natural number $n , I _ { n } - I _ { n + 1 } = \frac { \left( \frac { 1 } { 2 } \right) ^ { n + 1 } } { n + 1 }$.\\
b. Propose an algorithm to determine, for a given natural number $n$, the value of $I _ { n }$.
  \item Let $n$ be a non-zero natural number.
\end{enumerate}

It is admitted that if $x$ belongs to the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$ then $0 \leqslant \frac { x ^ { n } } { 1 - x } \leqslant \frac { 1 } { 2 ^ { n - 1 } }$.\\
a. Show that for every non-zero natural number $n$, $0 \leqslant I _ { n } \leqslant \frac { 1 } { 2 ^ { n } }$.\\
b. Deduce the limit of the sequence ( $I _ { n }$ ) as $n$ tends to $+ \infty$.\\
5. For every non-zero natural number $n$, we set

$$S _ { n } = \frac { 1 } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 } } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 3 } } { 3 } + \ldots + \frac { \left( \frac { 1 } { 2 } \right) ^ { n } } { n }$$

a. Show that for every non-zero natural number $n$, $S _ { n } = I _ { 0 } - I _ { n }$.\\
b. Determine the limit of $S _ { n }$ as $n$ tends to $+ \infty$.