Exercise 3 The purpose of this exercise is to study the convergence of two sequences towards the same limit. Part A Consider the function $f$ defined on $[2;+\infty[$ by $$f(x) = \sqrt{3x-2}$$
Justify the elements of the variation table below:
$x$
2
$+\infty$
$+\infty$
$f(x)$
2
We admit that the sequence $(u_n)$ satisfying $u_0 = 6$ and, for all natural number $n$, $u_{n+1} = f(u_n)$ is well defined.
[a.] Prove by induction that, for all natural number $n$: $2 \leqslant u_{n+1} \leqslant u_n \leqslant 6$.
[b.] Deduce that the sequence $(u_n)$ converges.
We call $\ell$ the limit of $(u_n)$. We admit that it is a solution of the equation $f(x) = x$. Determine the value of $\ell$.
Consider the rank function written below in Python language. We recall that $\operatorname{sqrt}(x)$ returns the square root of the number $x$. \begin{verbatim} from math import * def rang(a) : u=6 n=0 while u >= a : u = sqrt(3*u - 2) n = n+1 return n \end{verbatim}
[a.] Why can we affirm that rang(2.000001) returns a value?
[b.] For which values of the parameter $a$ does the instruction rang($a$) return a result?
Part B We admit that the sequence $(v_n)$ satisfying $v_0 = 6$ and, for all natural number $n$, $v_{n+1} = 3 - \dfrac{2}{v_n}$ is well defined.
Calculate $v_1$.
For all natural number $n$, we admit that $v_n \neq 2$ and we set: $$w_n = \frac{v_n - 1}{v_n - 2}$$
[a.] Prove that the sequence $(w_n)$ is geometric with ratio 2 and specify its first term $w_0$.
[b.] We admit that, for all natural number $n$, $$w_n - 1 = \frac{1}{v_n - 2}$$ Deduce that, for all natural number $n$, $$v_n = 2 + \frac{1}{1{,}25 \times 2^n - 1}$$
[c.] Calculate the limit of $(v_n)$.
Determine the smallest natural number $n$ for which $v_n < 2{,}01$ by solving the inequality.
Part C Using the previous parts, determine the smallest integer $N$ such that for all $n \geqslant N$, the terms $v_n$ and $u_n$ belong to the interval $]1{,}99;2{,}01[$.
\textbf{Exercise 3}
The purpose of this exercise is to study the convergence of two sequences towards the same limit.
\textbf{Part A}
Consider the function $f$ defined on $[2;+\infty[$ by
$$f(x) = \sqrt{3x-2}$$
\begin{enumerate}
\item Justify the elements of the variation table below:
\begin{center}
\begin{tabular}{|c|lll|}
\hline
$x$ & 2 & $+\infty$ & \\
\hline
& & $+\infty$ & \\
$f(x)$ & & & \\
& 2 & & \\
\hline
\end{tabular}
\end{center}
We admit that the sequence $(u_n)$ satisfying $u_0 = 6$ and, for all natural number $n$, $u_{n+1} = f(u_n)$ is well defined.
\item \begin{enumerate}
\item[a.] Prove by induction that, for all natural number $n$: $2 \leqslant u_{n+1} \leqslant u_n \leqslant 6$.
\item[b.] Deduce that the sequence $(u_n)$ converges.
\end{enumerate}
\item We call $\ell$ the limit of $(u_n)$.
We admit that it is a solution of the equation $f(x) = x$. Determine the value of $\ell$.
\item Consider the rank function written below in Python language.
We recall that $\operatorname{sqrt}(x)$ returns the square root of the number $x$.
\begin{verbatim}
from math import *
def rang(a) :
u=6
n=0
while u >= a :
u = sqrt(3*u - 2)
n = n+1
return n
\end{verbatim}
\begin{enumerate}
\item[a.] Why can we affirm that rang(2.000001) returns a value?
\item[b.] For which values of the parameter $a$ does the instruction rang($a$) return a result?
\end{enumerate}
\end{enumerate}
\textbf{Part B}
We admit that the sequence $(v_n)$ satisfying $v_0 = 6$ and, for all natural number $n$, $v_{n+1} = 3 - \dfrac{2}{v_n}$ is well defined.
\begin{enumerate}
\item Calculate $v_1$.
\item For all natural number $n$, we admit that $v_n \neq 2$ and we set:
$$w_n = \frac{v_n - 1}{v_n - 2}$$
\begin{enumerate}
\item[a.] Prove that the sequence $(w_n)$ is geometric with ratio 2 and specify its first term $w_0$.
\item[b.] We admit that, for all natural number $n$,
$$w_n - 1 = \frac{1}{v_n - 2}$$
Deduce that, for all natural number $n$,
$$v_n = 2 + \frac{1}{1{,}25 \times 2^n - 1}$$
\item[c.] Calculate the limit of $(v_n)$.
\end{enumerate}
\item Determine the smallest natural number $n$ for which $v_n < 2{,}01$ by solving the inequality.
\end{enumerate}
\textbf{Part C}
Using the previous parts, determine the smallest integer $N$ such that for all $n \geqslant N$, the terms $v_n$ and $u_n$ belong to the interval $]1{,}99;2{,}01[$.