Show that a function $P : \mathbb{Z} \rightarrow \mathbb{C}$ is quasi-polynomial if and only if there exist an integer $m \in \mathbb{N}^*$ and $m$ polynomials $P_0, \ldots, P_{m-1}$ with complex coefficients such that for all $j \in \{0, \ldots, m-1\}$ and for all $n \in \mathbb{Z}$ congruent to $j$ modulo $m$, we have $P(n) = P_j(n)$.

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

Show that for all $k \in \mathbf { N }$, the real numbers $b _ { k } = \sum _ { n = 1 } ^ { + \infty } \lambda _ { n } ^ { k } a _ { n }$ are well-defined.
The sequences satisfy: $\left| a _ { n } \right| \leq \frac { M } { 2 ^ { n } }$ for some $M \in \mathbf{R}_+^*$, and $\lambda_n$ is strictly increasing with $\lambda_0 = 0$, $\lim_{n\to+\infty} \lambda_n = +\infty$, and $\lambda_n \underset{n\to+\infty}{=} O(n)$.

Let $\omega$ be a root of unity and $p \in \mathbb{N}^*$. Let $\sum_{n=0}^{+\infty} R(n) x^n$ denote the power series expansion of $\frac{1}{(1 - \omega x)^p}$. Show that $R$ is a quasi-polynomial function then determine its degree and its leading coefficient.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\},$$ then we define the power series $F(x) = \sum_{n=0}^{\infty} P(n) x^n$.
Show that the radius of convergence of $F$ is greater than or equal to 1.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\},$$ then we define the power series $F(x) = \sum_{n=0}^{\infty} P(n) x^n$.
Prove the equality $F(x) = \prod_{i=1}^{k} \frac{1}{1 - x^{a_i}}$ for all $x \in ]-1, 1[$.

Let $k \in \mathbf { N } ^ { * }$. Show that $f \in \mathcal { C } ^ { k } \left( \mathbf { R } _ { + } , \mathbf { R } \right)$ and give an expression for $x \mapsto f ^ { ( k ) } ( x )$. Then express $f ^ { ( k ) } ( 0 )$ in terms of $b _ { k }$.
Here $f = \sum_{n\geq 0} f_n$ is the sum of a Dirichlet series with $f_n(x) = a_n e^{-\lambda_n x}$, and $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\},$$ then we define the power series $F(x) = \sum_{n=0}^{\infty} P(n) x^n$.
Deduce that $P$ is a quasi-polynomial function.

Show that if $f ( x ) = 0$ for all $x \in \mathbf { R } _ { + }$ then $a _ { n } = 0$ for all $n \in \mathbf { N }$.
Here $f = \sum_{n\geq 0} a_n e^{-\lambda_n x}$ is the sum of a Dirichlet series.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\},$$ then we define the power series $F(x) = \sum_{n=0}^{\infty} P(n) x^n$.
Calculate the leading coefficient of $P$.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$ We assume $k = 2$. We assume in this question that $(a_1, a_2) = (2, 3)$. Construct a function $\phi : \mathbb{Z} \rightarrow \mathbb{Z}$ of period 6 such that $P(n) = \frac{n + \phi(n)}{6}$ for all $n \in \mathbb{N}$.

Show that
$$\left\| y _ { N } - y \right\| _ { \infty , \mathbf { R } _ { + } } \leq \frac { M } { 2 ^ { N } }$$
and deduce that $y _ { N }$ converges uniformly to $y$ on $\mathbf { R } _ { + }$. Then propose an interval $J \subset \mathbf { R } _ { + }$ where the bound on $\left\| y _ { N } - y \right\| _ { \infty , J }$ would be sharper.
Here $y _ { N } ( x ) = \sum _ { n = 0 } ^ { N } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x }$ and $y ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x }$ with $\left| a_n \right| \leq \frac{M}{2^n}$.

By denoting $H _ { n } := \sum _ { k = 1 } ^ { n } \dfrac { 1 } { k }$ the harmonic series, show that $$H _ { n } \sim \ln n \quad ( n \rightarrow + \infty )$$

For all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, define $R _ { p , q } := \dfrac { 1 } { q } I _ { p , q }$ where $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x, \quad \alpha_{p,q} = \frac{p}{q}.$$
Using the change of variables $s = x ^ { n + 1 }$ in $I _ { p , q } ( n )$, prove that $$R _ { p , q } ( n ) \sim \frac { 1 } { 2 p n } \quad ( n \rightarrow + \infty )$$

For all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, define $R _ { p , q } := \dfrac { 1 } { q } I _ { p , q }$ where $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x, \quad \alpha_{p,q} = \frac{p}{q},$$ and recall that $$\phi _ { p , q } ( n ) = \frac { 1 } { q } \left( \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { \alpha _ { p , q } } } d x + ( - 1 ) ^ { n } I _ { p , q } ( n ) \right).$$
Using the result $R _ { p , q } ( n ) \sim \dfrac { 1 } { 2 p n }$ as $n \to +\infty$, deduce the convergence rate of the alternating congruent-harmonic series $\sum u _ { k }$, that is, that of the sequence of partial sums $\left( \phi _ { p , q } ( n ) \right) _ { n \in \mathbf { N } }$.

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
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = s + 1 } ^ { r } \frac { 1 } { k }$$

Show that, for all $t \in \mathbb{R}$, $$\sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{(t+h)x_i} = (2\operatorname{ch}(t+h))^n$$

We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We admit that $J _ { r , r } = \zeta ( 2 ) - \sum _ { k = 1 } ^ { r } \frac { 1 } { k ^ { 2 } }$.
Let $n \in \mathbb { N } ^ { * }$. Deduce that there exist two integers $p _ { n }$ and $q _ { n }$ such that
$$I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$$

Find the general term $T_r$ of the sequence $2, 7, 14, 23, 34, \ldots$

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

Find the integer part of $S = \displaystyle\sum_{k=2}^{9999} \dfrac{1}{\sqrt{k}}$.

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,
(A) $a _ { n } < b _ { n }$ for all $n > 1$
(B) $a _ { n } > b _ { n }$ for all $n > 1$
(C) $a _ { n } = b _ { n }$ for infinitely many $n$
(D) None of the above

For any $n \geq 5$, the value of $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2^n - 1}$ lies between
(A) 0 and $\frac{n}{2}$
(B) $\frac{n}{2}$ and $n$
(C) $n$ and $2n$
(D) none of the above.

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(A) $\frac{3}{4}$
(B) $\frac{1}{4}$
(C) 1
(D) 4