Show that there exists a real sequence $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ having the following property: for every integer $p \in \mathbb { N } ^ { * }$, for every non-degenerate interval $I$ and for every complex function $f$ of class $C ^ { \infty }$ on $I$, the function $g$ defined on $I$ by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { p - 1 } f ^ { ( p - 1 ) }$ satisfies $$g ^ { \prime } + \frac { 1 } { 2 ! } g ^ { \prime \prime } + \frac { 1 } { 3 ! } g ^ { ( 3 ) } + \cdots + \frac { 1 } { p ! } g ^ { ( p ) } = f ^ { \prime } + \sum _ { l = 1 } ^ { p - 1 } b _ { l , p } f ^ { ( p + l ) }$$ where the $b _ { l , p }$ are coefficients independent of $f$ which we do not seek to calculate.

Show that $a _ { 0 } = 1$ and that for every $p \geqslant 1 , a _ { p } = - \sum _ { i = 2 } ^ { p + 1 } \frac { a _ { p + 1 - i } } { i ! }$. Deduce that $\left| a _ { p } \right| \leqslant 1$ for every natural integer $p$. Determine $a _ { 1 }$ and $a _ { 2 }$.

a) For every $z \in \mathbb { C }$ such that $| z | < 1$, justify that the series $\sum _ { p \in \mathbb { N } } a _ { p } z ^ { p }$ is convergent.
We denote by $\varphi ( z )$ its sum: $\varphi ( z ) = \sum _ { p = 0 } ^ { \infty } a _ { p } z ^ { p }$.
b) For $z \in \mathbb { C }$ such that $| z | < 1$, calculate the product $\left( e ^ { z } - 1 \right) \varphi ( z )$. Deduce that for every $z \in \mathbb { C } ^ { * }$ satisfying $| z | < 1$, we have $\varphi ( z ) = \frac { z } { e ^ { z } - 1 }$.
c) Show that $a _ { 2 k + 1 } = 0$ for every integer $k \geqslant 1$. Calculate $a _ { 4 }$.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$.
Deduce the asymptotic expansion of the remainder $$R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } } = - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + a _ { 2 } f ^ { \prime \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right) + O \left( \frac { 1 } { n ^ { 2 p + \alpha - 1 } } \right)$$

Give the asymptotic expansion of $R _ { n } ( 3 )$ corresponding to the case $\alpha = 3$ and $p = 3$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } } = \left( A _ { n } ( X ) \right) _ { n \in \mathbb { N } }$ satisfying the following conditions $$A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n } \text { and } \int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0 \text { for every } n \in \mathbb { N }$$ The polynomials $B _ { n } = n ! A _ { n }$ are called Bernoulli polynomials.
a) Show that the sequence $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ is uniquely determined by the conditions above; specify the degree of $A _ { n }$; calculate $A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$.
b) Show that $A _ { n } ( t ) = ( - 1 ) ^ { n } A _ { n } ( 1 - t )$ for every $n \in \mathbb { N }$ and every $t \in \mathbb { R }$.
c) For every integer $n \geqslant 2$, show that $A _ { n } ( 0 ) = A _ { n } ( 1 )$ and that $A _ { 2 n - 1 } ( 0 ) = 0$.
d) We provisionally set $c _ { n } = A _ { n } ( 0 )$ for every natural integer $n$. Show that for every $n \in \mathbb { N }$, $$A _ { n } ( X ) = c _ { 0 } \frac { X ^ { n } } { n ! } + \cdots + c _ { n - 2 } \frac { X ^ { 2 } } { 2 ! } + c _ { n - 1 } X + c _ { n }$$ then that if $n \geqslant 1$, $$\frac { c _ { 0 } } { ( n + 1 ) ! } + \cdots + \frac { c _ { n - 2 } } { 3 ! } + \frac { c _ { n - 1 } } { 2 ! } + c _ { n } = 0$$
e) Deduce that for every $n \in \mathbb { N }$, we actually have $c _ { n } = a _ { n }$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ satisfying $A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n }$ and $\int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0$ for every $n \in \mathbb { N }$.
a) Show that the series $\sum _ { n } A _ { n } ( t ) z ^ { n }$ converges for every real $t \in [ - 1,1 ]$ and every complex $z$ satisfying $| z | < 1$.
Under these conditions, we set $f ( t , z ) = \sum _ { n = 0 } ^ { + \infty } A _ { n } ( t ) z ^ { n }$.
b) Let $z \in \mathbb { C }$ such that $| z | < 1$. Show that the function $t \mapsto f ( t , z )$ is differentiable on $[ 0,1 ]$ and express its derivative in terms of $f ( t , z )$. Deduce that if $| z | < 1$ and $z \neq 0$, $$\sum _ { n = 0 } ^ { + \infty } A _ { n } ( t ) z ^ { n } = \frac { z e ^ { t z } } { e ^ { z } - 1 }$$
c) Show that if $z \in \mathbb { C }$ and $| z | < 2 \pi$, we have $\frac { z e ^ { z / 2 } } { e ^ { z } - 1 } + \frac { z } { e ^ { z } - 1 } = 2 \frac { z / 2 } { e ^ { z / 2 } - 1 }$.
Deduce that for every natural integer $n , A _ { n } \left( \frac { 1 } { 2 } \right) = \left( \frac { 1 } { 2 ^ { n - 1 } } - 1 \right) a _ { n }$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ satisfying $A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n }$ and $\int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0$ for every $n \in \mathbb { N }$. For $n \geqslant 2$, the variations of $A_n$ on $[0,1]$ are known, in particular:

1. If $n \equiv 2 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } ( 1 ) > 0 > A _ { n } \left( \frac { 1 } { 2 } \right)$.
2. If $n \equiv 0 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } ( 1 ) < 0 < A _ { n } \left( \frac { 1 } { 2 } \right)$.
3. If $n \equiv 1 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } \left( \frac { 1 } { 2 } \right) = A _ { n } ( 1 ) = 0$; $A _ { n } < 0$ on $]0 , \frac{1}{2}[$ and $A _ { n } > 0$ on $]\frac{1}{2} , 1[$.
4. If $n \equiv 3 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } \left( \frac { 1 } { 2 } \right) = A _ { n } ( 1 ) = 0$; $A _ { n } > 0$ on $]0 , \frac{1}{2}[$ and $A _ { n } < 0$ on $]\frac{1}{2} , 1[$.

a) Show that for $n \geqslant 2$, the variations of the polynomials $A _ { n }$ on $[ 0,1 ]$ correspond to the four cases described above.
b) For every $n \in \mathbb { N } ^ { * }$ and every $x \in [ 0,1 ]$, show that $\left| A _ { 2 n } ( x ) \right| \leqslant \left| a _ { 2 n } \right|$ and $\left| A _ { 2 n + 1 } ( x ) \right| \leqslant \frac { \left| a _ { 2 n } \right| } { 2 }$.

Let $f$ be a complex function of class $C ^ { \infty }$ on $[ 0,1 ]$.
a) Show that for every integer $q \geqslant 1$ $$f ( 1 ) - f ( 0 ) = \sum _ { j = 1 } ^ { q } ( - 1 ) ^ { j + 1 } \left[ A _ { j } ( t ) f ^ { ( j ) } ( t ) \right] _ { 0 } ^ { 1 } + ( - 1 ) ^ { q } \int _ { 0 } ^ { 1 } A _ { q } ( t ) f ^ { ( q + 1 ) } ( t ) d t$$
b) Taking into account the relations found in the previous part, show that for every odd natural integer $q = 2 p + 1$ $$f ( 1 ) - f ( 0 ) = \frac { 1 } { 2 } \left( f ^ { \prime } ( 0 ) + f ^ { \prime } ( 1 ) \right) - \sum _ { j = 1 } ^ { p } a _ { 2 j } \left( f ^ { ( 2 j ) } ( 1 ) - f ^ { ( 2 j ) } ( 0 ) \right) - \int _ { 0 } ^ { 1 } A _ { 2 p + 1 } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$

Let $n \in \mathbb { N }$ and let $f$ be a real function of class $C ^ { \infty }$ on $[ n , + \infty [$. We assume that $f$ and all its derivatives have constant sign on $[ n , + \infty [$ and tend to 0 as $+ \infty$.
By applying, for $k \geqslant n$, the result of III.B.1 to $f _ { k } ( t ) = f ( k + t )$, show $$\sum _ { k = n } ^ { + \infty } f ^ { \prime } ( k ) = - f ( n ) + \frac { 1 } { 2 } f ^ { \prime } ( n ) - \sum _ { j = 1 } ^ { p } a _ { 2 j } f ^ { ( 2 j ) } ( n ) + \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$ where we have set $A _ { j } ^ { * } ( t ) = A _ { j } ( t - [ t ] )$ for every $t \in \mathbb { R }$.
Show that $$\left| \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t \right| \leqslant \left| \frac { a _ { 2 p } } { 2 } \right| \left| f ^ { ( 2 p + 1 ) } ( n ) \right|$$

Show that, in the expression of $R _ { n } ( \alpha )$ from II.B.2, the term $O \left( \frac { 1 } { n ^ { 2 p + \alpha - 1 } } \right)$ can be written in the form of an integral.

Let $g$ be a piecewise continuous increasing function on $[ 0,1 ]$.
By noting $\int _ { 0 } ^ { 1 } = \int _ { 0 } ^ { 1 / 2 } + \int _ { 1 / 2 } ^ { 1 }$, show that

• if $n \equiv 1 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \geqslant 0$;
• if $n \equiv 3 \bmod 4$, then $\int _ { 0 } ^ { 1 } A _ { n } ( t ) g ( t ) d t \leqslant 0$.

Using the notation from II.B.2, where $\widetilde { S } _ { n , 2 p - 2 } ( \alpha ) = \sum _ { k = 1 } ^ { n - 1 } \frac { 1 } { k ^ { \alpha } } - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right)$, show that for every natural integer $p \geqslant 1$ $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p + 2 } ( \alpha )$$ and that $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p - 2 } ( \alpha )$$ Deduce that the error $\left| S ( \alpha ) - \widetilde { S } _ { n , 2 p } ( \alpha ) \right|$ is bounded by $\left| a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) \right|$.

In this question, we return to the case of II.B.3 (i.e., $\alpha = 3$, $p = 3$, $n = 100$). Knowing that $6 ! a _ { 6 } = \frac { 1 } { 42 }$, recover that the error $\left| S ( 3 ) - \widetilde { S } _ { 100,4 } ( 3 ) \right|$ is bounded by an expression of order $10 ^ { - 17 }$.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Show that $\widetilde { A } _ { p }$ is $2 \pi$-periodic and piecewise continuous.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Using question III.B.1, determine the Fourier coefficients of $\widetilde { A } _ { p }$: $$\widehat { A } _ { p } ( n ) = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \widetilde { A } _ { p } ( t ) e ^ { - i n x } d x$$

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Study the convergence of the Fourier series of $\widetilde { A } _ { p }$.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. For $p \in \mathbb { N } ^ { * }$, deduce that $a _ { 2 p } = A _ { 2 p } ( 0 ) = ( - 1 ) ^ { p + 1 } \frac { S ( 2 p ) } { 2 ^ { 2 p - 1 } \pi ^ { 2 p } }$.

Show that, for all integers $n , p \geqslant 1$ $$\left| \frac { a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) } { a _ { 2 p } f ^ { ( 2 p ) } ( n ) } \right| = \frac { ( \alpha + 2 p ) ( \alpha + 2 p - 1 ) S ( 2 p + 2 ) } { 4 n ^ { 2 } \pi ^ { 2 } S ( 2 p ) }$$ where $f$ is defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$.

What can be said about the approximation of $S ( \alpha )$ by $\widetilde { S } _ { n , 2 p } ( \alpha )$ when, with $n$ fixed, $p$ tends to $+ \infty$ ? For the numerical calculation of $S ( \alpha )$, how should one choose $n$ and $p$ ?

For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ Justify the convergence of the series $\sum_{k \geqslant 2} w_{k}$.
Deduce that there exists a real number $a$ such that: $$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + v_{n}$$ where $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.

For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ and $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.
Deduce that $$\left| v_{n} - \frac{1}{12n} \right| \leqslant \frac{1}{12n^{2}}$$ then that: $$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + \frac{1}{12n} + O\left(\frac{1}{n^{2}}\right)$$

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Show that the function $H$ defined on $\mathbb{R}$ by: $$H(x) = \int_{0}^{x} h(t) \, dt$$ is continuous, of class $\mathcal{C}^{1}$ piecewise, and periodic with period 1.

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Is the application $u \longmapsto \dfrac{h(u)}{u+x}$ integrable on $\mathbb{R}_{+}$?

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Let $\varphi$ be the application defined for all $x > 0$ by: $$\varphi(x) = \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ By repeating the integration by parts from question IV.C, prove that the application $\varphi$ is of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{*}$ and that for all $x > 0$, $$\varphi^{\prime}(x) = -\int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$