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

2015 centrale-maths2__mp

44 maths questions

QI.A.1 Sequences and Series Convergence/Divergence Determination of Numerical Series View

Justify that the series with general term $a _ { n } = \frac { 1 } { n } - \int _ { n - 1 } ^ { n } \frac { \mathrm {~d} t } { t }$ converges.

QI.A.2 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$.
Show that there exists a real constant $A$ such that $H _ { n } \underset { + \infty } { = } \ln n + A + o ( 1 )$. Deduce that $H _ { n } \sim \ln n$.

QI.B Sequences and Series Convergence/Divergence Determination of Numerical Series View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$. We denote $\zeta$ the function defined for $x > 1$ by $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$.
Let $r$ be a natural integer. For which values of $r$ is the series $\sum _ { n \geqslant 1 } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ convergent?
In the rest of the problem we will denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.

QI.C.1 Sequences and Series Power Series Expansion and Radius of Convergence View

Give without proof the power series expansions of the functions $t \mapsto \ln ( 1 - t )$ and $t \mapsto \frac { 1 } { 1 - t }$ as well as their radius of convergence.

QI.C.2 Sequences and Series Power Series Expansion and Radius of Convergence View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$.
Deduce that the function $$t \mapsto - \frac { \ln ( 1 - t ) } { 1 - t }$$ is expandable as a power series on $] - 1,1 [$ and specify its power series expansion using the real numbers $H _ { n }$.

QI.D.1 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

QI.D.2 Integration by Parts Reduction Formula or Recurrence via Integration by Parts View

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Show that $\left. \forall p \in \mathbb { N } , \forall q \in \mathbb { N } ^ { * } , \forall \varepsilon \in \right] 0,1 \left[ , \quad I _ { p , q } ^ { \varepsilon } = - \frac { q } { p + 1 } I _ { p , q - 1 } ^ { \varepsilon } - \frac { \varepsilon ^ { p + 1 } ( \ln \varepsilon ) ^ { q } } { p + 1 } \right.$.

QI.D.3 Integration by Parts Reduction Formula or Recurrence via Integration by Parts View

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Deduce that we have $\forall p \in \mathbb { N } , \forall q \in \mathbb { N } ^ { * } , \quad I _ { p , q } = - \frac { q } { p + 1 } I _ { p , q - 1 }$.

QI.D.4 Integration by Parts Reduction Formula or Recurrence via Integration by Parts View

QI.E Integration by Parts Prove an Integral Identity or Equality View

Let $r$ be a non-zero natural integer and $f$ a function expandable as a power series on $] - 1,1 [$. We assume that for every $x$ in $] - 1,1 \left[ , f ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } \right.$ and that $\sum _ { n \geqslant 0 } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$ converges absolutely.
Show that $\int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } f ( t ) \mathrm { d } t = ( - 1 ) ^ { r - 1 } ( r - 1 ) ! \sum _ { n = 0 } ^ { + \infty } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$.

QI.F.1 Reduction Formulae Connect a Discrete Sum to an Integral via Reduction Formulae View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$ and $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.
Deduce from the previous questions that for every integer $r \geqslant 2$, $$S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } } = \frac { ( - 1 ) ^ { r } } { ( r - 1 ) ! } \int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } \frac { \ln ( 1 - t ) } { 1 - t } \mathrm {~d} t$$

QI.F.2 Reduction Formulae Connect a Discrete Sum to an Integral via Reduction Formulae View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$ and $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.
Establish that we then have $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \int _ { 0 } ^ { 1 } \frac { ( \ln t ) ^ { r - 2 } ( \ln ( 1 - t ) ) ^ { 2 } } { t } \mathrm {~d} t$.

QI.F.3 Reduction Formulae Connect a Discrete Sum to an Integral via Reduction Formulae View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$, $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges, and $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$.
Deduce that $S _ { 2 } = \frac { 1 } { 2 } \int _ { 0 } ^ { 1 } \frac { ( \ln t ) ^ { 2 } } { 1 - t } \mathrm {~d} t$ then find the value of $S _ { 2 }$ as a function of $\zeta ( 3 )$.

QII.A.1 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Let $x > 0$. Show that $t \mapsto t ^ { x - 1 } e ^ { - t }$ is integrable on $] 0 , + \infty [$.

QII.A.2 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Throughout the rest of the problem, we denote $\Gamma$ the function defined on $\mathbb { R } ^ { + * }$ by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that $\Gamma$ is of class $\mathcal { C } ^ { \infty }$ on its domain of definition, takes strictly positive values and satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.
Let $x$ and $\alpha$ be two strictly positive real numbers. Justify the existence of $\int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - \alpha t } \mathrm {~d} t$ and give its value as a function of $\Gamma ( x )$ and $\alpha ^ { x }$.

QII.B.1 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Justify the existence of $\beta ( x , y )$ for $x > 0$ and $y > 0$.

QII.B.2 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Show that for all real $x > 0$ and $y > 0 , \beta ( x , y ) = \beta ( y , x )$.

QII.B.3 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Let $x > 0$ and $y > 0$. Establish that $\beta ( x + 1 , y ) = \frac { x } { x + y } \beta ( x , y )$.

QII.B.4 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Deduce that for $x > 0 , y > 0 , \beta ( x + 1 , y + 1 ) = \frac { x y } { ( x + y ) ( x + y + 1 ) } \beta ( x , y )$.

QII.C.1 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View

QII.C.2 Integration by Substitution Substitution to Prove an Integral Identity or Equality View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$.
Show that $\beta ( x , y ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } \mathrm {~d} u$.
One may use the change of variable $t = \frac { u } { 1 + u }$.

QII.C.3 Reduction Formulae Bound or Estimate a Parametric Integral View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$.
We denote $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0. Show that $$\forall t \in \mathbb { R } ^ { + } , F _ { x , y } ( t ) \leqslant \Gamma ( x + y )$$

QII.C.4 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0.
Let $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Show that $G$ is defined and continuous on $\mathbb { R } ^ { + }$.

QII.C.5 Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0, and $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Show that $\lim _ { a \rightarrow + \infty } G ( a ) = \Gamma ( x + y ) \beta ( x , y )$.

QII.C.6 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0, and $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Show that $G$ is of class $\mathcal { C } ^ { 1 }$ on every segment $[ c , d ]$ included in $\mathbb { R } ^ { + * }$, then that $G$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } ^ { + * }$.

QII.C.7 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$, $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0, and $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Express for $a > 0$, $G ^ { \prime } ( a )$ as a function of $\Gamma ( x ) , e ^ { - a }$ and $a ^ { y - 1 }$.

QII.C.8 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. We want to show that for $x > 0$ and $y > 0$, $$\beta ( x , y ) = \frac { \Gamma ( x ) \Gamma ( y ) } { \Gamma ( x + y ) }$$ which will be denoted $(\mathcal{R})$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $G ( a ) = \int _ { 0 } ^ { + \infty } \frac { u ^ { x - 1 } } { ( 1 + u ) ^ { x + y } } F _ { x , y } ( ( 1 + u ) a ) \mathrm { d } u$.
Deduce from the above the relation $(\mathcal{R})$.

QIII.A Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View

We define the function $\psi$ (called the digamma function) on $\mathbb { R } ^ { + * }$ as the derivative of $x \mapsto \ln ( \Gamma ( x ) )$. For every real $x > 0 , \psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. We admit that $\Gamma$ satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.
Show that for every real $x > 0 , \psi ( x + 1 ) - \psi ( x ) = \frac { 1 } { x }$.

QIII.B.1 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. We have the relation $(\mathcal{R})$: $\beta ( x , y ) = \frac { \Gamma ( x ) \Gamma ( y ) } { \Gamma ( x + y ) }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$.
From the relation $(\mathcal{R})$, justify that $\frac { \partial \beta } { \partial y }$ is defined on $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$.
Establish that for all real $x > 0$ and $y > 0 , \frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$.

QIII.B.2 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Let $x > 0$ be fixed. What is the monotonicity on $\mathbb { R } ^ { + * }$ of the function $y \mapsto \beta ( x , y )$?

QIII.B.3 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$.
Show that the function $\psi$ is increasing on $\mathbb { R } ^ { + * }$.

QIII.C.1 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, satisfying $\psi ( x + 1 ) - \psi ( x ) = \frac{1}{x}$ for all $x > 0$.
Show that for every real $x > - 1$ and for every integer $n \geqslant 1$ $$\psi ( 1 + x ) - \psi ( 1 ) = \psi ( n + x + 1 ) - \psi ( n + 1 ) + \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { k } - \frac { 1 } { k + x } \right)$$

QIII.C.2 Reduction Formulae Bound or Estimate a Parametric Integral View

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, and $\psi$ is increasing on $\mathbb{R}^{+*}$.
Let $n$ be an integer $\geqslant 2$ and $x$ a real $> - 1$. We set $p = E ( x ) + 1$, where $E ( x )$ denotes the integer part of $x$. Prove that $$0 \leqslant \psi ( n + x + 1 ) - \psi ( n ) \leqslant H _ { n + p } - H _ { n - 1 } \leqslant \frac { p + 1 } { n }$$

QIII.C.3 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, satisfying $\psi ( x + 1 ) - \psi ( x ) = \frac{1}{x}$ for all $x > 0$.
Deduce that, for every real $x > - 1$, $$\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$$

QIII.D.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

We denote $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. We denote $g$ the function defined on $[ - 1 , + \infty [$ by $$g ( x ) = \sum _ { n = 2 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$$
Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $[ - 1 , + \infty [$.
Specify in particular the value of $g ^ { ( k ) } ( 0 )$ as a function of $\zeta ( k + 1 )$ for every integer $k \geqslant 1$.

QIII.D.2 Sequences and Series Power Series Expansion and Radius of Convergence View

We denote $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. We denote $g$ the function defined on $[ - 1 , + \infty [$ by $$g ( x ) = \sum _ { n = 2 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$$
Show that for every integer $n$ and for every $x$ in $] - 1,1 [$ $$\left| g ( x ) - \sum _ { k = 0 } ^ { n } \frac { g ^ { ( k ) } ( 0 ) } { k ! } x ^ { k } \right| \leqslant \zeta ( 2 ) | x | ^ { n + 1 }$$
Show that $g$ is expandable as a power series on $] - 1,1 [$.

QIII.D.3 Sequences and Series Power Series Expansion and Radius of Convergence View

We denote $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$. We have shown that for every real $x > -1$, $\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$.
Prove that for every $x$ in $] - 1,1 [$, $$\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } ( - 1 ) ^ { n + 1 } \zeta ( n + 1 ) x ^ { n }$$

QIV.A Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \frac { \partial ^ { 2 } \beta } { \partial y ^ { 2 } } ( x , 1 )$, where $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$ and $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. We have $\frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$.
Justify that $B$ is defined on $\mathbb { R } ^ { + * }$.
Using the relation found in III.B.1, establish that for every real $x > 0$ $$x B ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$$
Deduce that $B$ is $\mathcal { C } ^ { \infty }$ on $\mathbb { R } ^ { + * }$.

QIV.B.1 Reduction Formulae Establish an Integral Identity or Representation View

QIV.B.2 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View

We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$.
Give without justification an expression, using an integral, of $B ^ { ( p ) } ( x )$, for every natural integer $p$ and every real $x > 0$.

QIV.B.3 Reduction Formulae Connect a Discrete Sum to an Integral via Reduction Formulae View

We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, and $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$. We have shown that $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \int _ { 0 } ^ { 1 } \frac { ( \ln t ) ^ { r - 2 } ( \ln ( 1 - t ) ) ^ { 2 } } { t } \mathrm {~d} t$.
Deduce that for every integer $r \geqslant 2 , S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$.

QIV.B.4 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$, and $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$. We have $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$.
Find again the value of $S _ { 2 }$ already calculated in I.F.3.

QIV.C.1 Sequences and Series Properties and Manipulation of Power Series or Formal Series View

We define $\varphi$ the function defined on $] - 1 , + \infty [$ by $\varphi ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$.
Show that $\varphi$ is $\mathcal { C } ^ { \infty }$ on its domain of definition and give for every natural integer $n \geqslant 2$ the value of $\varphi ^ { ( n ) } ( 0 )$ as a function of the successive derivatives of $\psi$ at the point 1.

QIV.C.2 Sequences and Series Evaluation of a Finite or Infinite Sum View

We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$, and $\varphi ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$. We have $x B(x) = \varphi(x)$ for $x > 0$, and $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$. We have also shown that $\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } ( - 1 ) ^ { n + 1 } \zeta ( n + 1 ) x ^ { n }$ for $x \in ] -1, 1[$.
Conclude that, for every integer $r \geqslant 3$, $$2 S _ { r } = r \zeta ( r + 1 ) - \sum _ { k = 1 } ^ { r - 2 } \zeta ( k + 1 ) \zeta ( r - k )$$