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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

2013 centrale-maths1__pc

45 maths questions

QI.A.1 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a real decreasing sequence converging to 0, and $\left( b _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a complex sequence such that the sequence $\left( B _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ defined for all $n \in \mathbb { N } ^ { * }$ by $B _ { n } = b _ { 1 } + \cdots + b _ { n }$ is bounded.
Show that, for all integer $n \geqslant 2$, $$\sum _ { k = 1 } ^ { n } a _ { k } b _ { k } = a _ { n } B _ { n } + \sum _ { k = 1 } ^ { n - 1 } \left( a _ { k } - a _ { k + 1 } \right) B _ { k }$$

QI.A.2 Sequences and Series Convergence/Divergence Determination of Numerical Series View

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a real decreasing sequence converging to 0, and $\left( b _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a complex sequence such that the sequence $\left( B _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ defined for all $n \in \mathbb { N } ^ { * }$ by $B _ { n } = b _ { 1 } + \cdots + b _ { n }$ is bounded.
Deduce that the series $\sum a _ { n } b _ { n }$ converges.

QI.A.3 Sequences and Series Convergence/Divergence Determination of Numerical Series View

Show that, for all $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$, the series $\sum _ { n \geq 1 } \frac { e ^ { \mathrm { i } n \theta } } { n }$ converges.

QI.B.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

We consider the function series $\sum _ { n \geq 1 } \frac { \sin ( n x ) } { \sqrt { n } }$, where $x$ is a real variable.
Show that this function series converges pointwise on $\mathbb { R }$.

QI.B.2 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

We consider the function series $\sum _ { n \geq 1 } \frac { \sin ( n x ) } { \sqrt { n } }$, where $x$ is a real variable.
Show that it cannot be the Fourier series of a $2 \pi$-periodic piecewise continuous function.
One may begin by recalling Parseval's formula.

QI.C.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Show that $p$ is well defined, continuous and $2 \pi$-periodic.

QI.C.2 Sequences and Series Functional Equations and Identities via Series View

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Determine the Fourier series of $p$.

QI.C.3 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Show that the function $p$ is not of class $\mathcal { C } ^ { 1 }$.

QII.A.1 Sequences and Series Power Series Expansion and Radius of Convergence View

Let $\theta \in \mathbb { R }$.
Determine the radius of convergence of the power series $\sum \frac { e ^ { \mathrm { i n } \theta } } { n } x ^ { n }$.

QII.A.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View

Let $\theta \in \mathbb { R }$. Let $g$ be the function from $] - 1,1 [$ to $\mathbb { C }$ defined by $$g ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \mathrm { e } ^ { \mathrm { i } n \theta } } { n } x ^ { n }$$
a) Show that $g$ is of class $C ^ { 1 }$ on $] - 1,1 [$ and that, for all $x \in ] - 1,1 [$, $$g ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \mathrm { i } \theta } - x } { x ^ { 2 } - 2 x \cos \theta + 1 }$$
b) Show that, if $x \in ] - 1,1 [$, $$h ( x ) = - \frac { 1 } { 2 } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) + \mathrm { i } \arctan \left( \frac { x \sin \theta } { 1 - x \cos \theta } \right)$$ is well defined and that $h ( x ) = g ( x )$.

QII.B.1 Sequences and Series Functional Equations and Identities via Series View

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Show that, for all $n \in \mathbb { N } ^ { * }$, $$\sum _ { k = 1 } ^ { n } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { \mathrm { i } \theta } \frac { 1 - \left( \mathrm { e } ^ { \mathrm { i } \theta } t \right) ^ { n } } { 1 - \mathrm { e } ^ { \mathrm { i } \theta } t } \mathrm { ~d} t$$

QII.B.2 Sequences and Series Evaluation of a Finite or Infinite Sum View

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Deduce that $$\sum _ { k = 1 } ^ { + \infty } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { \mathrm { i } \theta } } { 1 - \mathrm { e } ^ { \mathrm { i } \theta } t } \mathrm { ~d} t$$
One may use the dominated convergence theorem.

QII.B.3 Sequences and Series Evaluation of a Finite or Infinite Sum View

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Deduce that $$\sum _ { k = 1 } ^ { + \infty } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = - \frac { 1 } { 2 } \ln ( 2 - 2 \cos \theta ) + \mathrm { i } \arctan \left( \frac { \sin \theta } { 1 - \cos \theta } \right)$$

QII.B.4 Sequences and Series Evaluation of a Finite or Infinite Sum View

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Show that, for all $\theta \in ] 0 , \pi [$, $$\sum _ { k = 1 } ^ { + \infty } \frac { \sin ( k \theta ) } { k } = \frac { \pi - \theta } { 2 }$$

QII.C.1 Sequences and Series Functional Equations and Identities via Series View

Let $r : \mathbb { R } \rightarrow \mathbb { R }$, be a $2 \pi$-periodic, odd function, such that $\forall \theta \in ] 0 , \pi ] , r ( \theta ) = \frac { \pi - \theta } { 2 }$.
Justify the existence and uniqueness of $r$.

QII.C.2 Sequences and Series Functional Equations and Identities via Series View

Let $r : \mathbb { R } \rightarrow \mathbb { R }$, be a $2 \pi$-periodic, odd function, such that $\forall \theta \in ] 0 , \pi ] , r ( \theta ) = \frac { \pi - \theta } { 2 }$.
Determine the Fourier series of $r$.

QII.C.3 Sequences and Series Evaluation of a Finite or Infinite Sum View

Let $r : \mathbb { R } \rightarrow \mathbb { R }$, be a $2 \pi$-periodic, odd function, such that $\forall \theta \in ] 0 , \pi ] , r ( \theta ) = \frac { \pi - \theta } { 2 }$.
Deduce that $\sum _ { n = 0 } ^ { + \infty } \frac { 1 } { ( 2 n + 1 ) ^ { 2 } } = \frac { \pi ^ { 2 } } { 8 }$.

QIII.A.1 Inequalities Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

Show that if $x$ is a real number different from 1 and from $-1$, then $x ^ { 2 } - 2 x \cos \theta + 1 > 0$ for all $\theta \in \mathbb { R }$.

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

Study the convergence of the improper integrals $$\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 - \cos \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 + \cos \theta ) d \theta$$
Deduce that, for all $x \in \mathbb { R }$, the integral $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ converges.

QIII.A.3 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Show that, as $x$ tends to $+ \infty$, $$2 \pi \ln ( x ) - \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$$ has a limit, which one will determine.

QIII.A.4 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Show that $x \mapsto \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ is an even function of the variable $x \in \mathbb { R }$.

QIII.B.1 Sequences and Series Functional Equations and Identities via Series View

Let $x \in ] - 1,1 [$.
Determine the Fourier series of the function $\widetilde { h } : \mathbb { R } \rightarrow \mathbb { R }$ defined by $\widetilde { h } ( \theta ) = \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right)$.
One may use the result from question II.A.2.

QIII.B.2 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Deduce that, for all $x \in ] - 1,1 [$, we have $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = 0$.
Deduce the value of $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ in the case $| x | > 1$.

QIII.B.3 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Show that the improper integral $\int _ { 0 } ^ { \pi / 2 } \ln ( \cos \theta ) d \theta$ converges.

QIII.B.4 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Show that $\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) \mathrm { d } \theta = 2 \int _ { 0 } ^ { \pi / 2 } \ln ( \sin \theta ) \mathrm { d } \theta = 2 \int _ { 0 } ^ { \pi / 2 } \ln ( \cos \theta ) \mathrm { d } \theta$.

QIII.B.5 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Deduce that $\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) \mathrm { d } \theta = - \pi \ln 2$.

QIII.B.6 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Deduce that $\int _ { 0 } ^ { \pi } \ln ( 2 - 2 \cos \theta ) \mathrm { d } \theta = \int _ { 0 } ^ { \pi } \ln ( 2 + 2 \cos \theta ) \mathrm { d } \theta = 0$.

QIII.C.1 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is differentiable on $\mathbb { R } \backslash \{ - 1,1 \}$ and that $$\forall x \in \mathbb { R } \backslash \{ - 1,1 \} \quad f ^ { \prime } ( x ) = \int _ { 0 } ^ { \pi } \frac { 2 x - 2 \cos \theta } { x ^ { 2 } - 2 x \cos \theta + 1 } \mathrm { ~d} \theta$$

QIII.C.2 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$ $$f ^ { \prime } ( x ) = 4 \int _ { 0 } ^ { + \infty } \frac { ( x + 1 ) t ^ { 2 } + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } t ^ { 2 } + ( x - 1 ) ^ { 2 } \right) \left( t ^ { 2 } + 1 \right) } \mathrm { d } t$$

QIII.C.3 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $$f ( x ) = \begin{cases} 2 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$
One will first determine coefficients $A$ and $B$ as functions of $x$ such that $\frac { ( x + 1 ) T + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } \right) ( T + 1 ) } = \frac { A } { ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } } + \frac { B } { T + 1 }$ for all $T \in \mathbb { R }$ such that these fractions are defined.

QIII.C.4 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is continuous on $\mathbb { R }$ and that $f ( 1 ) = f ( - 1 ) = 0$.
One may show that $\forall x \in \mathbb { R } , x ^ { 2 } - 2 x \cos \theta + 1 \geqslant \sin ^ { 2 } \theta$ and use the dominated convergence theorem.

QIII.D.1 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

Show that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$ $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = \lim _ { n \rightarrow + \infty } \left( \frac { 2 \pi } { n } \sum _ { k = 1 } ^ { n } \ln \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) \right)$$

QIII.D.2 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View

Show that, for all $x \in \mathbb { R }$ and for all $n \in \mathbb { N } ^ { * }$, $$\left( x ^ { n } - 1 \right) ^ { 2 } = \prod _ { k = 1 } ^ { n } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right)$$

QIII.D.3 Reduction Formulae Establish an Integral Identity or Representation View

Deduce that $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) d \theta = \begin{cases} 4 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$

QIII.D.4 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Deduce $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ for $x \in \mathbb { R } \backslash \{ - 1,1 \}$.

QIII.D.5 Sequences and Series Functional Equations and Identities via Series View

Show that $\forall x \in \mathbb { R }$ and $\forall n \in \mathbb { N } ^ { * }$ $$\prod _ { k = 1 } ^ { n - 1 } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) = \left( \sum _ { k = 0 } ^ { n - 1 } x ^ { k } \right) ^ { 2 }$$

QIII.D.6 Sequences and Series Evaluation of a Finite or Infinite Sum View

Show that $\prod _ { k = 1 } ^ { n - 1 } \sin \frac { k \pi } { 2 n } = \frac { \sqrt { n } } { 2 ^ { n - 1 } }$.

QIII.D.7 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Deduce that $$\int _ { 0 } ^ { \pi / 2 } \ln ( \sin \theta ) d \theta = - \pi \frac { \ln 2 } { 2 }$$
Then recover the result from question III.B.6.

QIV.A.1 Sequences and series, recurrence and convergence Series convergence and power series analysis View

QIV.A.2 Sequences and series, recurrence and convergence Series convergence and power series analysis View

QIV.A.3 Sequences and series, recurrence and convergence Series convergence and power series analysis View

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a complex sequence. We assume that the series $\sum a _ { n }$ converges. For $n \in \mathbb { N }$, we denote $r _ { n } = \sum _ { k = n + 1 } ^ { + \infty } a _ { k }$ and we define the functions $s _ { n }$ and $s$ from $[ 0,1 ]$ to $\mathbb { C }$ by $s _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }$ and $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$.
Show that $s$ is continuous on $[ 0,1 ]$.
For continuity at 1, fix $\varepsilon > 0$ and show that if the natural integer $N$ satisfies $\left| r _ { n } \right| \leqslant \varepsilon$ for all $n \geqslant N$, then $\left| s ( x ) - s _ { N } ( x ) \right| \leqslant 2 \varepsilon$ for all $x \in [ 0,1 ]$. Then bound the modulus of $s ( x ) - s ( 1 ) = \left( s ( x ) - s _ { N } ( x ) \right) + \left( s _ { N } ( x ) - s _ { N } ( 1 ) \right) + \left( s _ { N } ( 1 ) - s ( 1 ) \right)$.

QIV.A.4 Sequences and series, recurrence and convergence Series convergence and power series analysis View

QIV.B Sequences and series, recurrence and convergence Series convergence and power series analysis View

Let $\theta \in \mathbb { R }$. Determine the power series expansion of the function $$x \mapsto \frac { 1 - x ^ { 2 } } { x ^ { 2 } - 2 x \cos \theta + 1 }$$ on an interval to be specified.

QIV.C.1 Sequences and series, recurrence and convergence Series convergence and power series analysis View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a $2 \pi$-periodic function of class $\mathcal { C } ^ { 1 }$. We consider the Fourier series of $f$ in cosines and sines, denoted $$c _ { 0 } + \sum _ { n \geq 1 } \left( a _ { n } \cos ( n t ) + b _ { n } \sin ( n t ) \right)$$
Show that, for all $x \in ] - 1,1 [$ and all $t \in \mathbb { R }$, $$c _ { 0 } + \sum _ { n = 1 } ^ { + \infty } \left( a _ { n } \cos ( n t ) + b _ { n } \sin ( n t ) \right) x ^ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \frac { \left( 1 - x ^ { 2 } \right) f ( u ) } { x ^ { 2 } - 2 x \cos ( t - u ) + 1 } \mathrm { ~d} u$$

QIV.C.2 Sequences and series, recurrence and convergence Convergence proof and limit determination View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a $2 \pi$-periodic function of class $\mathcal { C } ^ { 1 }$. We consider the Fourier series of $f$ in cosines and sines, denoted $$c _ { 0 } + \sum _ { n \geq 1 } \left( a _ { n } \cos ( n t ) + b _ { n } \sin ( n t ) \right)$$
Deduce that, for all $t \in \mathbb { R }$, $$f ( t ) = \lim _ { x \rightarrow 1 ^ { - } } \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \frac { \left( 1 - x ^ { 2 } \right) f ( u ) } { x ^ { 2 } - 2 x \cos ( t - u ) + 1 } \mathrm { ~d} u$$