grandes-ecoles

Papers (191)

2025

centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38

2024

centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26

2023

centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15

2022

centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23

2021

centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44

2020

centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18

2019

centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26

2018

centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24

2017

centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26

2016

centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20

2015

centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22

2014

centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24

2013

centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9

2012

centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18

2011

centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10

2010

centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27

2022 centrale-maths2__pc

39 maths questions

Q1 Proof Proof That a Map Has a Specific Property View

Show that $\langle \cdot , \cdot \rangle$ is an inner product on $\mathbb { R } _ { n - 1 } [ X ]$, where $$\langle P , Q \rangle = \sum _ { k = 1 } ^ { n } P \left( a _ { k } \right) Q \left( a _ { k } \right).$$

Q2 Proof Direct Proof of a Stated Identity or Equality View

Show that, for all $i$ and $k$ in $\llbracket 1 , n \rrbracket$, $$L _ { i } \left( a _ { k } \right) = \begin{cases} 1 & \text { if } k = i \\ 0 & \text { otherwise } \end{cases}$$ where $L_i(X) = \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \frac { X - a _ { j } } { a _ { i } - a _ { j } }$.

Q3 Proof Direct Proof of a Stated Identity or Equality View

Show that, for all $i \in \llbracket 1 , n \rrbracket$ and all $P \in \mathbb { R } _ { n - 1 } [ X ]$, $$\left\langle L _ { i } , P \right\rangle = P \left( a _ { i } \right).$$

Q5 Proof Deduction or Consequence from Prior Results View

Deduce that, for all $P \in \mathbb { R } _ { n - 1 } [ X ]$, $$P = \sum _ { i = 1 } ^ { n } P \left( a _ { i } \right) L _ { i }.$$

Q6 Proof Direct Proof of a Stated Identity or Equality View

Show that, for any polynomial $P$ of degree at most $n - 2$, $$\sum _ { i = 1 } ^ { n } \frac { P \left( a _ { i } \right) } { \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \left( a _ { i } - a _ { j } \right) } = 0 .$$

Q7 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ By expanding $( 1 + x ) ^ { n }$ for two appropriately chosen real numbers $x$, show that $$\sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } \binom { n } { 2 p } = 2 ^ { n - 1 }.$$

Q8 Factor & Remainder Theorem Polynomial Degree and Structural Properties View

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ Show that $T _ { n }$ is a polynomial of degree $n$. Explicitly state the leading coefficient of $T _ { n }$.

Q9 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ Show that $T _ { n }$ is the unique polynomial with real coefficients satisfying the relation $$\forall \theta \in \mathbb { R } , \quad T _ { n } ( \cos ( \theta ) ) = \cos ( n \theta ).$$

Q10 Roots of polynomials Factored form and root structure from polynomial identities View

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ For $k \in \llbracket 1 , n \rrbracket$, we set $y _ { k , n } = \cos \left( \frac { ( 2 k - 1 ) \pi } { 2 n } \right)$. Show that $$T _ { n } ( X ) = 2 ^ { n - 1 } \prod _ { k = 1 } ^ { n } \left( X - y _ { k , n } \right).$$

Q11 Roots of polynomials Proof of polynomial identity or inequality involving roots View

Let $n \in \mathbb{N}^*$ and $W$ be a monic polynomial of degree $n$. The objective of this subsection is to show that $$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }.$$ Show that $\sup _ { x \in [ - 1,1 ] } \left| T _ { n } ( x ) \right| = 1$. Deduce a monic polynomial of degree $n$ achieving equality in the above inequality.

Q12 Factor & Remainder Theorem Polynomial Degree and Structural Properties View

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, and set $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$. Show that $Q$ is a polynomial of degree at most $n - 1$.

Q13 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

If we assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | < \frac { 1 } { 2 ^ { n - 1 } }$, show that, for all $k \in \llbracket 0 , n - 1 \rrbracket , Q \left( z _ { k } \right) Q \left( z _ { k + 1 } \right) < 0$.
Deduce a contradiction and conclude.

Q14 Proof Direct Proof of an Inequality View

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$, and for all $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. We now assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | = \frac { 1 } { 2 ^ { n - 1 } }$. Show that, for all $k \in \llbracket 0 , n \rrbracket$, $$\frac { Q \left( z _ { k } \right) } { \prod _ { \substack { j = 0 \\ j \neq k } } ^ { n } \left( z _ { k } - z _ { j } \right) } \geqslant 0.$$

Q15 Proof Deduction or Consequence from Prior Results View

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$, and for all $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. We assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | = \frac { 1 } { 2 ^ { n - 1 } }$. Deduce that $Q = 0$, then that $W = \frac { 1 } { 2 ^ { n - 1 } } T _ { n }$.
One may consider the sum of the inequalities from the previous question and exploit question 6 applied to suitable data.

Q16 Proof Existence Proof View

Let $r$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I = [a,b]$ and vanishing at $n + 1$ distinct points of $I$. Show that there exists $c \in I$ such that $r ^ { ( n ) } ( c ) = 0$.

Q17 Proof Existence Proof View

Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $W = \prod_{i=1}^n (X - a_i)$ and let $f$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I$. Let $P = \Pi ( f )$ be the interpolation polynomial of $f$ associated with the real numbers $a _ { 1 } , \ldots , a _ { n }$, defined by $$\Pi ( f ) = \sum _ { i = 1 } ^ { n } f \left( a _ { i } \right) L _ { i }.$$ For all $x \in I$, show that there exists $c \in I$ such that $$f ( x ) - P ( x ) = \frac { f ^ { ( n ) } ( c ) } { n ! } W ( x ).$$ For $x$ distinct from the $a _ { i }$, one may consider the function $r$ defined on $I$ by $$r ( t ) = f ( t ) - P ( t ) - K W ( t )$$ where the real number $K$ is chosen so that $r ( x ) = 0$.

Q18 Taylor series Lagrange error bound application View

Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $f$ be a real-valued function of class $\mathcal{C}^n$ on $I$ and $P = \Pi(f)$ its Lagrange interpolation polynomial. Deduce that $$\sup _ { x \in [ a , b ] } | f ( x ) - P ( x ) | \leqslant \frac { M _ { n } ( b - a ) ^ { n } } { n ! }$$ where $M _ { n } = \sup _ { x \in [ a , b ] } \left| f ^ { ( n ) } ( x ) \right|$.

Q19 Taylor series Lagrange error bound application View

Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. For all $n \in \mathbb { N } ^ { * }$, let $P _ { n } = \Pi _ { n } ( f )$ be the Lagrange interpolation polynomial of $f$ at $n$ distinct points $a_{1,n} < \cdots < a_{n,n}$ of $I$. Show that the sequence $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $I$.

Q20 Sequences and series, recurrence and convergence Sequence of functions convergence View

Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. Show that there exists a sequence of polynomials $\left( Q _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ that converges uniformly towards $f$ on $I$ and such that, for all $n \in \mathbb { N } ^ { * }$, the function $Q _ { n }$ does not coincide with $f$ at any point of $I$, except possibly at zero: $$\forall n \in \mathbb { N } ^ { * } , \quad \forall x \in I \backslash \{ 0 \} , \quad Q _ { n } ( x ) \neq \exp ( x ).$$

Q21 Taylor series Prove smoothness or power series expandability of a function View

Let $a > 0$, $I = [-a, a]$, and $$f : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{1+x^2} \end{array}.$$ Show that $f$ is of class $\mathcal { C } ^ { \infty }$ and that, for all $k$ in $\mathbb { N }$ and all $t \in \left] - \pi / 2 , \pi / 2 \right[$, $$f ^ { ( k ) } ( \tan t ) = k ! \cos ^ { k + 1 } ( t ) \cos ( ( k + 1 ) t + k \pi / 2 ).$$

Q22 Taylor series Lagrange error bound application View

Let $a > 0$, $I = [-a, a]$, and $f(x) = \dfrac{1}{1+x^2}$ for $x \in \mathbb{R}$. For all $n \in \mathbb { N } ^ { * }$, let $P _ { n } = \Pi _ { n } ( f )$ be the Lagrange interpolation polynomial of $f$ on $I$. Show that, if $a < \frac { 1 } { 2 }$, the sequence of polynomials $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $[ - a , a ]$.

Q23 Taylor series Derive series via differentiation or integration of a known series View

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$. We set $$\forall x \in ] - 1,1 [ , \quad g ( x ) = \sum _ { k = 0 } ^ { + \infty } x ^ { k }.$$ Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $] - 1,1 [$ and that $$\forall j \in \mathbb { N } , \quad \forall x \in ] - 1,1 [ , \quad g ^ { ( j ) } ( x ) = \frac { j ! } { ( 1 - x ) ^ { j + 1 } }.$$

Q24 Taylor series Determine radius or interval of convergence View

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$ and $r \in ] 0 , R [$. Show that there exists $C \in \mathbb { R }$ such that $$\forall k \in \mathbb { N } , \quad \left| c _ { k } \right| \leqslant \frac { C } { r ^ { k } }.$$

Q25 Taylor series Lagrange error bound application View

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$, $r \in ]0, R[$, and $f(x) = \sum_{k=0}^{+\infty} c_k x^k$ for $x \in ]-R, R[$. Let $C \in \mathbb{R}$ be such that $|c_k| \leq C/r^k$ for all $k \in \mathbb{N}$. Deduce that for all $x \in ] - r , r [$ and for all $n \in \mathbb { N }$, $$\left| f ^ { ( n ) } ( x ) \right| \leqslant \frac { n ! r C } { ( r - | x | ) ^ { n + 1 } }.$$

Q26 Taylor series Lagrange error bound application View

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$, $f(x) = \sum_{k=0}^{+\infty} c_k x^k$ for $x \in ]-R, R[$, and $a > 0$. Assume that $a < R / 3$. Show that the sequence of polynomials $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } } = \left( \Pi _ { n } ( f ) \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $[ - a , a ]$.

Q27 Taylor series Lagrange error bound application View

Let $a > 0$, $I = [-a, a]$. For all $n \in \mathbb { N } ^ { * }$, the Chebyshev points of order $n$ in $I$ are $$a _ { k , n } ^ { * } = a \cos \left( \frac { ( 2 k - 1 ) \pi } { 2 n } \right) , \quad \text { for } k \in \llbracket 1 , n \rrbracket ,$$ and $W _ { n } ^ { * } ( X ) = \prod _ { k = 1 } ^ { n } \left( X - a _ { k , n } ^ { * } \right)$. For all $x \in [ - a , a ]$, show that $\left| W _ { n } ^ { * } ( x ) \right| \leqslant 2 \left( \frac { a } { 2 } \right) ^ { n }$.

Q28 Taylor series Lagrange error bound application View

Let $a > 0$, $I = [-a,a]$, and $f(x) = \dfrac{1}{1+x^2}$ for $x \in \mathbb{R}$. For all $n \in \mathbb{N}^*$, the Chebyshev points of order $n$ in $I$ are $a_{k,n}^* = a\cos\left(\frac{(2k-1)\pi}{2n}\right)$ for $k \in \llbracket 1,n \rrbracket$, and $\Pi_n^*(f)$ denotes the interpolation polynomial of $f$ at these points. Show that, if $a < 2$, the sequence $\left( \Pi _ { n } ^ { * } ( f ) \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $[ - a , a ]$.

Q29 Taylor series Lagrange error bound application View

Let $\sum_{k \geqslant 0} c_k x^k$ be a power series with radius of convergence $R > 0$, $f(x) = \sum_{k=0}^{+\infty} c_k x^k$ for $x \in ]-R,R[$, and $a > 0$. For all $n \in \mathbb{N}^*$, the Chebyshev points of order $n$ in $[-a,a]$ are $a_{k,n}^* = a\cos\left(\frac{(2k-1)\pi}{2n}\right)$ for $k \in \llbracket 1,n \rrbracket$, and $\Pi_n^*(f)$ denotes the interpolation polynomial of $f$ at these points. Show that, if $a < 2R/3$, the sequence $\left( \Pi _ { n } ^ { * } ( f ) \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $[ - a , a ]$.

Q30 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$. Show that $h _ { \alpha }$ is a continuous decreasing integrable function on $[ 0,1 [$.

Q31 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$ and set $J _ { \alpha } = \int _ { 0 } ^ { 1 } h _ { \alpha } ( t ) \, \mathrm { d } t$. Justify that $$J _ { \alpha } = \int _ { 0 } ^ { 1 } \ln ( 1 - t ) \, \mathrm { d } t + \int _ { 0 } ^ { 1 } \ln ( 1 + t ) \, \mathrm { d } t - \int _ { 0 } ^ { 1 } \ln \left( \alpha ^ { 2 } + t ^ { 2 } \right) \mathrm { d } t = \int _ { 0 } ^ { 2 } \ln ( u ) \, \mathrm { d } u - \int _ { 0 } ^ { 1 } \ln \left( \alpha ^ { 2 } + t ^ { 2 } \right) \mathrm { d } t.$$

Q32 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$ and set $J _ { \alpha } = \int _ { 0 } ^ { 1 } h _ { \alpha } ( t ) \, \mathrm { d } t$. Deduce that $$J _ { \alpha } = 2 \ln ( 2 ) - \ln \left( 1 + \alpha ^ { 2 } \right) - 2 \alpha \arctan \left( \frac { 1 } { \alpha } \right).$$

Q33 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

For any real $\alpha > 0$, consider $J _ { \alpha } = 2 \ln ( 2 ) - \ln \left( 1 + \alpha ^ { 2 } \right) - 2 \alpha \arctan \left( \frac { 1 } { \alpha } \right)$. Show that there exists $\gamma > 0$ such that, for all $\alpha \in ] 0 , \gamma [$, $J _ { \alpha } > 0$.

Q34 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n }$ given, for $k \in \llbracket 0 , n - 1 \rrbracket$, by $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ and $$S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right).$$ For all $n \in \mathbb { N } ^ { * }$, show that $$\int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t + \frac { 1 } { n } h _ { \alpha } \left( \frac { 2 n - 1 } { 2 n } \right) \leqslant S _ { n } \left( h _ { \alpha } \right) \leqslant \frac { 1 } { n } h _ { \alpha } \left( \frac { 1 } { 2 n } \right) + \int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t.$$

Q35 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ for $k \in \llbracket 0 , n - 1 \rrbracket$, and $S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right)$, where $J_\alpha = \int_0^1 h_\alpha(t)\,\mathrm{d}t$. Deduce that the sequence $\left( S _ { n } \left( h _ { \alpha } \right) \right) _ { n \in \mathbb { N } ^ { * } }$ converges to $J _ { \alpha }$.

Q36 Sequences and Series Limit Evaluation Involving Sequences View

For all $n \in \mathbb { N } ^ { * }$, consider the points $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ for $k \in \llbracket 0 , n - 1 \rrbracket$. Let $\gamma > 0$ be such that $J_\alpha > 0$ for all $\alpha \in ]0, \gamma[$. Show that, for $\alpha \in ] 0 , \gamma [$, the sequence $\left( \left| \prod _ { k = 0 } ^ { n - 1 } \frac { 1 - a _ { k , n } ^ { 2 } } { \alpha ^ { 2 } + a _ { k , n } ^ { 2 } } \right| \right) _ { n \in \mathbb { N } ^ { * } }$ diverges to $+ \infty$.

Q37 Polynomial Division & Manipulation View

Let $I = [ - 1,1 ]$, $\alpha > 0$, and $$f _ { \alpha } : \begin{array}{ccc} {[-1,1]} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{\alpha^2 + x^2} \end{array}.$$ For $n \in \mathbb{N}^*$, let $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, and let $R_n \in \mathbb{R}_{2n-1}[X]$ be the interpolating polynomial of $f_\alpha$ at the $2n$ real numbers $\{\pm a_{k,n} \mid k \in \llbracket 0, n-1 \rrbracket\}$. Set $Q_n(X) = 1 - (X^2 + \alpha^2) R_n(X)$. Show that $R_n$ is an even polynomial and determine $Q_n(\alpha \mathrm{i})$.

Q38 Polynomial Division & Manipulation View

Let $I = [-1,1]$, $\alpha > 0$, $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, $R_n \in \mathbb{R}_{2n-1}[X]$ the interpolating polynomial of $f_\alpha(x) = \frac{1}{\alpha^2+x^2}$ at $\{\pm a_{k,n} \mid k \in \llbracket 0,n-1\rrbracket\}$, and $Q_n(X) = 1 - (X^2 + \alpha^2)R_n(X)$. Show that there exists $\lambda_n \in \mathbb{R}$ such that $$\forall x \in [ - 1,1 ] , \quad Q _ { n } ( x ) = \lambda _ { n } \prod _ { k = 0 } ^ { n - 1 } \left( x ^ { 2 } - a _ { k , n } ^ { 2 } \right).$$

Q39 Polynomial Division & Manipulation View

Let $I = [-1,1]$, $\alpha > 0$, $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, $R_n \in \mathbb{R}_{2n-1}[X]$ the interpolating polynomial of $f_\alpha(x) = \frac{1}{\alpha^2+x^2}$ at $\{\pm a_{k,n} \mid k \in \llbracket 0,n-1\rrbracket\}$, and $Q_n(X) = \lambda_n \prod_{k=0}^{n-1}(x^2 - a_{k,n}^2)$. Deduce that, for all $x \in [ - 1,1 ]$, $$f _ { \alpha } ( x ) - R _ { n } ( x ) = \frac { ( - 1 ) ^ { n } } { x ^ { 2 } + \alpha ^ { 2 } } \prod _ { k = 0 } ^ { n - 1 } \frac { x ^ { 2 } - a _ { k , n } ^ { 2 } } { \alpha ^ { 2 } + a _ { k , n } ^ { 2 } }.$$

Q40 Sequences and Series Limit Evaluation Involving Sequences View

Let $I = [-1,1]$, $\alpha > 0$, $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, $R_n \in \mathbb{R}_{2n-1}[X]$ the interpolating polynomial of $f_\alpha(x) = \frac{1}{\alpha^2+x^2}$ at $\{\pm a_{k,n} \mid k \in \llbracket 0,n-1\rrbracket\}$, and $\gamma > 0$ such that $J_\alpha > 0$ for all $\alpha \in ]0,\gamma[$. Suppose that $\alpha < \gamma$. Show that $$\lim _ { n \rightarrow + \infty } \left| f _ { \alpha } ( 1 ) - R _ { n } ( 1 ) \right| = + \infty.$$