For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)!\zeta(2n)}{2^{2n-1}\pi^{2n}}$$ Using the relation $\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = 0$ for $n \geqslant 1$, calculate $b_2$, $b_4$ and $b_6$, then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.

Suppose (for this question only) that $h$ is the function $x \mapsto x$. Determine a solution of $(E_{h})$: $$\forall x \in \mathbb{K},\, f(x+1) - f(x) = x$$ in $\mathbb{K}_{2}[X]$, then all polynomial solutions of the equation $(E_{h})$.

Using the Bernoulli polynomials $(B_n)_{n \in \mathbb{N}}$ satisfying $B_n(z+1) - B_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$, deduce the expression of a polynomial function satisfying the equation $(E_h)$: $$\forall x \in \mathbb{C},\, f(x+1) - f(x) = h(x)$$ on $\mathbb{C}$ when $h$ is a polynomial function.

Recall that $x$ is a fixed element of $]0;1[$. Deduce that:
$$\frac { \pi } { \sin ( \pi x ) } = \frac { 1 } { x } - \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } x } { n ^ { 2 } - x ^ { 2 } }$$

Show that for all $n \in \mathbf { N } ^ { * }$:
$$\int _ { \frac { \pi } { 2 } + ( n - 1 ) \pi } ^ { \frac { \pi } { 2 } + n \pi } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \cos ( t ) ) ^ { 2 p } \frac { 2 ( - 1 ) ^ { n } t \sin ( t ) } { t ^ { 2 } - n ^ { 2 } \pi ^ { 2 } } \mathrm {~d} t$$

Deduce that:
$$\int _ { \frac { \pi } { 2 } } ^ { + \infty } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \cos ( t ) ) ^ { 2 p } \left( \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } t \sin ( t ) } { t ^ { 2 } - n ^ { 2 } \pi ^ { 2 } } \right) \mathrm { d } t$$

Let $G = ( S , A )$ be a graph with $| S | = n \geq 2$. We write $\chi _ { G } ( X ) = X ^ { n } + \sum _ { k = 0 } ^ { n - 1 } a _ { k } X ^ { k }$. Give the value of $a _ { n - 1 }$ and express $a _ { n - 2 }$ in terms of $| A |$.

Show the equality $$v_n(k) = \sum_{i=1}^{r} b_i e^{a_i} \int_{a_i}^{\infty} e^{-t} t^{n-kr} (t - a_1)^k \cdots (t - a_r)^k\, dt.$$

Show that for any non-zero natural integer $n$, $$D_{n} = n! \sum_{k=0}^{n} \frac{(-1)^{k}}{k!}$$

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_{n}$ such that $\omega(\sigma) = k$.
Show that $$\frac{1}{n!} \sum_{k=1}^{n} k(k-1) s(n,k) = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^{2}}.$$

Let $\left(a_{n}\right)_{n \geqslant 2}$ be a sequence of real numbers. For $t \in \mathbb{R}$, we set $A(t) = \sum_{2 \leqslant k \leqslant t} a_{k}$. Let $b : [2, +\infty[ \rightarrow \mathbb{R}$ be a function of class $\mathscr{C}^{1}$. Show that for any integer $n \geqslant 2$, $$\sum_{k=2}^{n} a_{k} b(k) = A(n) b(n) - \int_{2}^{n} b^{\prime}(t) A(t) \mathrm{d}t.$$

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Let $\left(a_n\right)_{n \geqslant 2}$ be a sequence of real numbers. For $t \in \mathbb{R}$, we set $A(t) = \sum_{2 \leqslant k \leqslant t} a_k$. Let $b : [2, +\infty[ \rightarrow \mathbb{R}$ be a function of class $\mathscr{C}^1$. Show that for any integer $n \geqslant 2$, $$\sum_{k=2}^{n} a_k b(k) = A(n)b(n) - \int_2^n b'(t) A(t) \, \mathrm{d}t$$

Prove that $$\sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { 2 n + 1 } = \frac { \pi } { 4 }$$

Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \sum _ { k = 0 } ^ { + \infty } \frac { 1 } { ( r + k + 1 ) ( s + k + 1 ) }$$
Deduce that
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = 0 } ^ { + \infty } \left( \frac { 1 } { s + k + 1 } - \frac { 1 } { r + k + 1 } \right)$$

Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = 0 } ^ { + \infty } \left( \frac { 1 } { s + k + 1 } - \frac { 1 } { r + k + 1 } \right)$$
Deduce that
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = s + 1 } ^ { r } \frac { 1 } { k }$$

In this question, we set $p = q = 1$. Show that $$\phi _ { 1,1 } ( n ) = \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + t } d t - \int _ { 0 } ^ { 1 } \frac { ( - t ) ^ { n + 1 } } { 1 + t } d t$$ where $\phi_{1,1}(n) = \sum_{k=0}^{n} \dfrac{(-1)^k}{k+1}$.

Deduce the value of $S _ { 1,1 }$, the sum of the congruent-harmonic series with parameters $p=q=1$.

Show that, for all $q \geq 2$, $$S _ { 1 , q } = ( - 1 ) ^ { q } \left( \phi _ { 1,1 } ( q - 2 ) - \ln 2 \right)$$

113. We classify the natural numbers in groups such that each group contains consecutive numbers, i.e., $\ldots, \{4,5,6\}, \{2,3\}, \{1\}, \ldots$ What is the sum of the numbers in the group containing 350?
$$4125 \quad (1) \qquad 4050 \quad (2) \qquad 4015 \quad (3) \qquad 3980 \quad (4)$$

Find $\lim_{n \to \infty} u_n$ where $u_n = \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \cdots + \dfrac{n}{2^n}$.

The value of $$1 + \frac { 1 } { 1 + 2 } + \frac { 1 } { 1 + 2 + 3 } + \cdots + \frac { 1 } { 1 + 2 + 3 + \cdots 2021 }$$ is
(A) $\frac { 2021 } { 1010 }$.
(B) $\frac { 2021 } { 1011 }$.
(C) $\frac { 2021 } { 1012 }$.
(D) $\frac { 2021 } { 1013 }$.

The value of $\cot \left( \sum _ { n = 1 } ^ { 23 } \cot ^ { - 1 } \left( 1 + \sum _ { k = 1 } ^ { n } 2 k \right) \right)$ is
(A) $\frac { 23 } { 25 }$
(B) $\frac { 25 } { 23 }$
(C) $\frac { 23 } { 24 }$
(D) $\frac { 24 } { 23 }$

Let $S _ { n } = \sum _ { k = 1 } ^ { 4 n } ( - 1 ) ^ { \frac { k ( k + 1 ) } { 2 } } k ^ { 2 }$. Then $S _ { n }$ can take value(s)
(A) 1056
(B) 1088
(C) 1120
(D) 1332

The sum of the series $1 ^ { 2 } + 2.2 ^ { 2 } + 3 ^ { 2 } + 2.4 ^ { 2 } + 5 ^ { 2 } + 2.6 ^ { 2 } + \ldots . + 2 ( 2 m ) ^ { 2 }$ is
(1) $m ( 2 m + 1 ) ^ { 2 }$
(2) $m ^ { 2 } ( m + 2 )$
(3) $m ^ { 2 } ( 2 m + 1 )$
(4) $m ( m + 2 ) ^ { 2 }$

The sum of the series $1 + \frac { 4 } { 3 } + \frac { 10 } { 9 } + \frac { 28 } { 27 } + \ldots$ upto $n$ terms is
(1) $\frac { 7 } { 6 } n + \frac { 1 } { 6 } - \frac { 2 } { 3.2 ^ { n - 1 } }$
(2) $\frac { 5 } { 3 } n - \frac { 7 } { 6 } + \frac { 1 } { 2.3 ^ { n - 1 } }$
(3) $n + \frac { 1 } { 2 } - \frac { 1 } { 2 \cdot 3 ^ { n } }$
(4) $n - \frac { 1 } { 3 } - \frac { 1 } { 3.2 ^ { n - 1 } }$