The objective of this exercise is to conjecture in Part A and then prove in Part B the behavior of a sequence. The two parts can, however, be treated independently. We consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 3$ and for all $n \in \mathbb { N }$ :
$$u _ { n + 1 } = \frac { 4 } { 5 - u _ { n } } .$$
Part A - Copy and complete the following Python function suite(n) which takes as parameter the rank $n$ and returns the value of the term $u _ { n }$.
\begin{verbatim} def suite(n): u = ... for i in range(n) : ... return u \end{verbatim}
\setcounter{enumi}{1} - The execution of suite(2) returns 1.3333333333333333 .
Perform a calculation to verify and explain this output.
3. Using the outputs below, make a conjecture about the direction of variation and a conjecture about the convergence of the sequence ( $u _ { n }$ ).
\begin{verbatim} > suite(2) 1.3333333333333333 >> suite(5) 1.0058479532163742 >> suite(10) 1.0000057220349845 > suite(20) 1.000000000005457 \end{verbatim}
Part B We consider the function $f$ defined and differentiable on the interval $] - \infty ; 5 [$ by:
$$f ( x ) = \frac { 4 } { 5 - x }$$
Thus, the sequence ( $u _ { n }$ ) is defined by $u _ { 0 } = 3$ and for all $n \in \mathbb { N } , u _ { n + 1 } = f \left( u _ { n } \right)$.
- Show that the function $f$ is increasing on the interval $] - \infty$; 5[.
- Prove by induction that for every natural integer $n$ we have:
$$1 \leqslant u _ { n + 1 } \leqslant u _ { n } \leqslant 4 .$$
\setcounter{enumi}{2} - a. Let $x$ be a real number in the interval $] - \infty$; 5[. Prove the following equivalence:
$$f ( x ) = x \Longleftrightarrow x ^ { 2 } - 5 x + 4 = 0 .$$
b. Solve $f ( x ) = x$ in the interval $] - \infty$; 5[. - Prove that the sequence ( $u _ { n }$ ) is convergent. Determine its limit.
- Would the behavior of the sequence be identical by choosing as initial term $u _ { 0 } = 4$ instead of $u _ { 0 } = 3$ ?