Verify that
$$\begin{cases} I _ { n , k } = 0 & \text { if } n > k \text { or if } k - n \text { is odd } \\ I _ { n , k } = \frac { ( - 1 ) ^ { p } } { 2 ^ { n + 2 p } } \binom { n + 2 p } { n + p } & \text { if } k = n + 2 p \text { with } p \geqslant 0 \end{cases}$$

Deduce the power series development, for $n \geqslant 0$ and $x \in \mathbb { R }$ :
$$\varphi _ { n } ( x ) = \sum _ { p = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { p } } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Specify the radius of convergence.

Show that $\varphi _ { n }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbb { R }$.

We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
From which value $p _ { 0 }$ of $p$ is the sequence $\left( a _ { p } \right) _ { p \in \mathbb { N } }$ decreasing?

We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Assume $N > p _ { 0 }$. Bound $\left| R _ { N } \right|$ as a function of $(N, n, x)$ with
$$R _ { N } = \sum _ { p = N + 1 } ^ { + \infty } ( - 1 ) ^ { p } a _ { p }$$
Deduce, for fixed $\varepsilon > 0$, a sufficient condition on $N$ for $\left| \varphi _ { n } ( x ) - S _ { N } \right| < \varepsilon$.

We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Write a Maple or Mathematica function, CalculPhi, with arguments $(n, x, \varepsilon)$ returning an approximate value of $\varphi _ { n } ( x )$ to within $\varepsilon$. The coefficients $a _ { p }$ will be calculated by recursion.

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $M_n = \sum_{k=1}^n \dfrac{1}{2\cos\dfrac{k\pi}{2n+1}}$.
Give an equivalent of $M_n$ as $n$ tends to $+\infty$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. From the previous questions, there exist $a_n \in \mathbb{C}$ such that $c_{n,f}(r) = a_n r^{|n|}$ for all $r \in \mathbb{R}_+^*$.
By stating precisely the theorem used, establish $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = \lim _ { p \rightarrow \infty } \sum _ { n = - p } ^ { p } a _ { n } r ^ { | n | } e ^ { i n \theta }$$

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 $n \geqslant 2$, $$\sum _ { k = 1 } ^ { n } a _ { k } b _ { k } = a _ { n } B _ { n } + \sum _ { k = 1 } ^ { n - 1 } \left( a _ { k } - a _ { k + 1 } \right) B _ { k }$$

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.

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.

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 $\mathbb { R }$.

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.

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.

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 $p$.

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 $\mathcal { C } ^ { 1 }$.

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

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 $h ( x ) = g ( x )$.

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

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.

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

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

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 $r$.

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 $r$.

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 $\sum _ { n = 0 } ^ { + \infty } \frac { 1 } { ( 2 n + 1 ) ^ { 2 } } = \frac { \pi ^ { 2 } } { 8 }$.