a) Prove that, for every natural integer $n$, $u_{n+1} + u_{n} = \frac{1}{n+1}$. b) Deduce the exact value of $u_{1}$.
a) Copy and complete the algorithm below so that it displays as output the term of rank $n$ of the sequence $(u_{n})$ where $n$ is a natural integer entered as input by the user.
Variables :
$i$ and $n$ are natural integers, $u$ is a real number
Input :
Enter $n$
Initialization :
Assign to $u$ the value ...
Processing :
\begin{tabular}{l} For $i$ varying from 1 to... | Assign to $u$ the value . . .
End For
\hline & \hline Output : & Display $u$ \hline \end{tabular} b) Using this algorithm, the following table of values was obtained:
$n$
0
1
2
3
4
5
10
50
100
$u_{n}$
0,6931
0,3069
0,1931
0,1402
0,1098
0,0902
0,0475
0,0099
0,0050
What conjectures concerning the behavior of the sequence $(u_{n})$ can be made?
a) Prove that the sequence $(u_{n})$ is decreasing. b) Prove that the sequence $(u_{n})$ is convergent.
We call $\ell$ the limit of the sequence $(u_{n})$. Prove that $\ell = 0$.
We define the sequence $\left(u_{n}\right)$ as follows: for every natural integer $n$, $u_{n} = \int_{0}^{1} \frac{x^{n}}{1+x} \mathrm{~d}x$.
\begin{enumerate}
\item Calculate $u_{0} = \int_{0}^{1} \frac{1}{1+x} \mathrm{~d}x$.
\item a) Prove that, for every natural integer $n$, $u_{n+1} + u_{n} = \frac{1}{n+1}$.\\
b) Deduce the exact value of $u_{1}$.
\item a) Copy and complete the algorithm below so that it displays as output the term of rank $n$ of the sequence $(u_{n})$ where $n$ is a natural integer entered as input by the user.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Variables : & $i$ and $n$ are natural integers, $u$ is a real number \\
\hline
Input : & Enter $n$ \\
\hline
Initialization : & Assign to $u$ the value ... \\
\hline
Processing : & \begin{tabular}{l}
For $i$ varying from 1 to... | Assign to $u$ the value . . . \\
End For \\
\end{tabular} \\
\hline
& \\
\hline
Output : & Display $u$ \\
\hline
\end{tabular}
\end{center}
b) Using this algorithm, the following table of values was obtained:
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
$n$ & 0 & 1 & 2 & 3 & 4 & 5 & 10 & 50 & 100 \\
\hline
$u_{n}$ & 0,6931 & 0,3069 & 0,1931 & 0,1402 & 0,1098 & 0,0902 & 0,0475 & 0,0099 & 0,0050 \\
\hline
\end{tabular}
\end{center}
What conjectures concerning the behavior of the sequence $(u_{n})$ can be made?
\item a) Prove that the sequence $(u_{n})$ is decreasing.\\
b) Prove that the sequence $(u_{n})$ is convergent.
\item We call $\ell$ the limit of the sequence $(u_{n})$. Prove that $\ell = 0$.
\end{enumerate}