We consider the function $f$ defined on the interval $[0;+\infty[$ by $$f(x) = 5 - \frac{4}{x+2}$$ It will be admitted that $f$ is differentiable on the interval $[0;+\infty[$. The curve $\mathscr{C}$ representing $f$ and the line $\mathscr{D}$ with equation $y = x$ have been drawn in an orthonormal coordinate system in Appendix 1.
Prove that $f$ is increasing on the interval $[0;+\infty[$.
Solve the equation $f(x) = x$ on the interval $[0;+\infty[$. We denote the solution by $\alpha$. The exact value of $\alpha$ will be given, then an approximate value to $10^{-2}$ will be given.
We consider the sequence $(u_{n})$ defined by $u_{0} = 1$ and, for every natural integer $n$, $u_{n+1} = f(u_{n})$. On the figure in Appendix 1, using the curve $\mathscr{C}$ and the line $\mathscr{D}$, place the points $M_{0}$, $M_{1}$ and $M_{2}$ with zero ordinate and abscissae $u_{0}$, $u_{1}$ and $u_{2}$ respectively. What conjectures can be made about the direction of variation and the convergence of the sequence $(u_{n})$?
a. Prove, by induction, that for every natural integer $n$, $$0 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha$$ where $\alpha$ is the real number defined in question 2. b. Can we affirm that the sequence $(u_{n})$ is convergent? The answer will be justified.
For every natural integer $n$, we define the sequence $(S_{n})$ by $$S_{n} = \sum_{k=0}^{n} u_{k} = u_{0} + u_{1} + \cdots + u_{n}$$ a. Calculate $S_{0}$, $S_{1}$ and $S_{2}$. Give an approximate value of the results to $10^{-2}$ near. b. Complete the algorithm given in Appendix 2 so that it displays the sum $S_{n}$ for the value of the integer $n$ requested from the user. c. Show that the sequence $(S_{n})$ diverges to $+\infty$.
We consider the function $f$ defined on the interval $[0;+\infty[$ by
$$f(x) = 5 - \frac{4}{x+2}$$
It will be admitted that $f$ is differentiable on the interval $[0;+\infty[$. The curve $\mathscr{C}$ representing $f$ and the line $\mathscr{D}$ with equation $y = x$ have been drawn in an orthonormal coordinate system in Appendix 1.
\begin{enumerate}
\item Prove that $f$ is increasing on the interval $[0;+\infty[$.
\item Solve the equation $f(x) = x$ on the interval $[0;+\infty[$. We denote the solution by $\alpha$. The exact value of $\alpha$ will be given, then an approximate value to $10^{-2}$ will be given.
\item We consider the sequence $(u_{n})$ defined by $u_{0} = 1$ and, for every natural integer $n$, $u_{n+1} = f(u_{n})$.
On the figure in Appendix 1, using the curve $\mathscr{C}$ and the line $\mathscr{D}$, place the points $M_{0}$, $M_{1}$ and $M_{2}$ with zero ordinate and abscissae $u_{0}$, $u_{1}$ and $u_{2}$ respectively.\\
What conjectures can be made about the direction of variation and the convergence of the sequence $(u_{n})$?
\item a. Prove, by induction, that for every natural integer $n$,
$$0 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha$$
where $\alpha$ is the real number defined in question 2.\\
b. Can we affirm that the sequence $(u_{n})$ is convergent? The answer will be justified.
\item For every natural integer $n$, we define the sequence $(S_{n})$ by
$$S_{n} = \sum_{k=0}^{n} u_{k} = u_{0} + u_{1} + \cdots + u_{n}$$
a. Calculate $S_{0}$, $S_{1}$ and $S_{2}$. Give an approximate value of the results to $10^{-2}$ near.\\
b. Complete the algorithm given in Appendix 2 so that it displays the sum $S_{n}$ for the value of the integer $n$ requested from the user.\\
c. Show that the sequence $(S_{n})$ diverges to $+\infty$.
\end{enumerate}