grandes-ecoles 2024 QSpec-I

grandes-ecoles · France · geipi-polytech__maths 20 marks Sequences and series, recurrence and convergence Direct term computation from recurrence
Mathematics Specialty - EXERCISE I (20 points)

First Part
Consider the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ defined by $u _ { 0 } = 2$ and for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined for every positive real $x$ by $f ( x ) = \frac { 3 x + 2 } { x + 4 }$. We admit that, for every natural number $\boldsymbol { n }$, $\boldsymbol { u } _ { \boldsymbol { n } }$ is greater than or equal to $1$.
I-1-a- Calculate the exact values of $u _ { 1 }$ and $u _ { 2 }$. Give the result as an irreducible fraction.
I-1-b- The graph below gives the representative curve in an orthonormal coordinate system of the function $f$. From this graph, what can be conjectured about the variations and convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$? Specify the possible limit.
Second Part - Method 1
I-2-a- Show that, for every natural number $n$, $u _ { n + 1 } - u _ { n } = \frac { \left( 1 - u _ { n } \right) \left( u _ { n } + 2 \right) } { u _ { n } + 4 }$. I-2-b- Deduce the direction of variation of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Justify your answer. I-3- Prove that the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is convergent. Let $l$ denote its limit. I-4- Determine the value of $l$. Justify your answer.
Third Part - Method 2
Consider the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ defined for every natural number $n$ by: $v _ { n } = \frac { u _ { n } - 1 } { u _ { n } + 2 }$. I-5- Calculate $v _ { 0 }$. I-6-a- Determine the constant $k$ in $] 0 ; 1 [$ such that $v _ { n + 1 } = k \times v _ { n }$ for every natural number $n$. Justify your answer. What can be deduced about the nature of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$? For questions $\mathbf { I - 6 - b }$ and $\mathbf { I - 6 - c }$, answers may be expressed as a function of $k$ or its value. I-6-b- Deduce the expression of $v _ { n }$ as a function of $n$. I-6-c- Deduce the convergence of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer. I-7-a- Express $u _ { n }$ as a function of $v _ { n }$ for every natural number $n$. I-7-b- Deduce the convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer. 
\section*{Mathematics Specialty - EXERCISE I (20 points)}

\section*{First Part}
Consider the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ defined by $u _ { 0 } = 2$ and for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined for every positive real $x$ by $f ( x ) = \frac { 3 x + 2 } { x + 4 }$.\\
We admit that, for every natural number $\boldsymbol { n }$, $\boldsymbol { u } _ { \boldsymbol { n } }$ is greater than or equal to $1$.

I-1-a- Calculate the exact values of $u _ { 1 }$ and $u _ { 2 }$. Give the result as an irreducible fraction.

I-1-b- The graph below gives the representative curve in an orthonormal coordinate system of the function $f$. From this graph, what can be conjectured about the variations and convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$? Specify the possible limit.

\section*{Second Part - Method 1}
I-2-a- Show that, for every natural number $n$, $u _ { n + 1 } - u _ { n } = \frac { \left( 1 - u _ { n } \right) \left( u _ { n } + 2 \right) } { u _ { n } + 4 }$.\\
I-2-b- Deduce the direction of variation of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Justify your answer.\\
I-3- Prove that the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is convergent. Let $l$ denote its limit.\\
I-4- Determine the value of $l$. Justify your answer.

\section*{Third Part - Method 2}
Consider the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ defined for every natural number $n$ by: $v _ { n } = \frac { u _ { n } - 1 } { u _ { n } + 2 }$.\\
I-5- Calculate $v _ { 0 }$.\\
I-6-a- Determine the constant $k$ in $] 0 ; 1 [$ such that $v _ { n + 1 } = k \times v _ { n }$ for every natural number $n$. Justify your answer. What can be deduced about the nature of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$?\\
For questions $\mathbf { I - 6 - b }$ and $\mathbf { I - 6 - c }$, answers may be expressed as a function of $k$ or its value.\\
I-6-b- Deduce the expression of $v _ { n }$ as a function of $n$.\\
I-6-c- Deduce the convergence of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer.\\
I-7-a- Express $u _ { n }$ as a function of $v _ { n }$ for every natural number $n$.\\
I-7-b- Deduce the convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer.
Paper Questions
QI QII QIII QIV QSpec-I QSpec-II QV QVI QVII