For all integers $n, k \geq 0$, define the rational number $v_n(k)$ as the coefficient of degree $n$ in the power series $$\left(1 - s_1 x - \cdots - s_r x^r\right)^k v(x) = \sum_{n=0}^{\infty} v_n(k) x^n.$$ Show that $v(x)$ is the power series expansion of a rational fraction if and only if there exists an integer $k \geq 0$ such that $\sum_{n=0}^{\infty} v_n(k) x^n$ is a polynomial.

For all integers $n, k \geq 0$, define the rational number $v_n(k)$ as the coefficient of degree $n$ in the power series $$\left(1 - s_1 x - \cdots - s_r x^r\right)^k v(x) = \sum_{n=0}^{\infty} v_n(k) x^n.$$ Observe the equality: for all $n \geq r$ and $k \geq 0$, $$v_n(k+1) = v_n(k) - s_1 v_{n-1}(k) - \cdots - s_r v_{n-r}(k).$$

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that if $f$ is a function $E$, then the numerical series $\sum_{n=0}^{\infty} \frac{b_n}{n!} \alpha^n$ converges for every real number $\alpha$. We denote its value by $f(\alpha)$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f$ be a function $E$ that is not a polynomial. Show that there exists $R > 0$ such that the numerical series $\sum_{n=0}^{\infty} b_n \alpha^n$ diverges for every real number $\alpha$ with $|\alpha| > R$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Recall that $\widehat{f}(x)$ denotes the Laplace transform $\widehat{f}(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} b_n x^n$. Which functions $E$ are such that $\widehat{f}$ is also a function $E$?

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Prove that functions $E$ are closed under addition and multiplication.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f$ be an exponential polynomial (i.e., $f(x) = \sum_{i=1}^{s} P_i(x) e^{c_i x}$ with $c_i \in \mathbf{Q}$ and $P_i \in \mathbf{Q}[x]$). Show that $f$ is a function $E$ such that $\widehat{f}$ is the power series expansion of a rational fraction with rational poles.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that if $\sum_{n=0}^{\infty} b_n x^n$ is the power series expansion of a rational fraction with rational poles, then $\sum_{n=0}^{\infty} \frac{b_n}{n!} x^n$ is a function $E$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that the Bessel function $$J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$$ is a function $E$ such that $\widehat{J}_0(x)$ satisfies the equation $(1 + x^2)\widehat{J}_0(x)^2 = 1$. Deduce that $J_0(x)$ is not an exponential polynomial.

Show that the real zeros of the Bessel function $J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$ are simple, that is, if $J_0(\alpha) = 0$, then $J_0'(\alpha) \neq 0$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f(x)$ be a function $E$ such that $f(1) = 0$. Show that the power series $f(x)/(x-1)$ is still a function $E$.

We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
Prove that, for all $n$ in $\mathbb { N } ^ { * } , \left| b _ { n } \right| \leqslant 1$. Deduce an inequality on the radius of convergence of the power series $\sum _ { k \geqslant 0 } b _ { k } t ^ { k }$.

Show that every Dirichlet series $\sum _ { n \geq 0 } f _ { n }$ converges uniformly on $\mathbf { R } _ { + }$. We then denote its sum by $f$. Justify that $f$ is continuous on $\mathbf { R } _ { + }$.
A Dirichlet series satisfies $f_n(x) = a_n e^{-\lambda_n x}$ with $\left| a _ { n } \right| \leq \frac { M } { 2 ^ { n } }$, $\lambda_0 = 0$, $\lim_{n\to+\infty}\lambda_n = +\infty$, and $\lambda_n = O(n)$.

For positive integer $n$, define
$$f ( n ) = n + \frac { 16 + 5 n - 3 n ^ { 2 } } { 4 n + 3 n ^ { 2 } } + \frac { 32 + n - 3 n ^ { 2 } } { 8 n + 3 n ^ { 2 } } + \frac { 48 - 3 n - 3 n ^ { 2 } } { 12 n + 3 n ^ { 2 } } + \cdots + \frac { 25 n - 7 n ^ { 2 } } { 7 n ^ { 2 } }$$
Then, the value of $\lim _ { n \rightarrow \infty } f ( n )$ is equal to
(A) $3 + \frac { 4 } { 3 } \log _ { e } 7$
(B) $4 - \frac { 3 } { 4 } \log _ { e } \left( \frac { 7 } { 3 } \right)$
(C) $4 - \frac { 4 } { 3 } \log _ { e } \left( \frac { 7 } { 3 } \right)$
(D) $3 + \frac { 3 } { 4 } \log _ { e } 7$

If $1 + \frac { \sqrt { 3 } - \sqrt { 2 } } { 2 \sqrt { 3 } } + \frac { 5 - 2 \sqrt { 6 } } { 18 } + \frac { 9 \sqrt { 3 } - 11 \sqrt { 2 } } { 36 \sqrt { 3 } } + \frac { 49 - 20 \sqrt { 6 } } { 180 } + \ldots$ upto $\infty = 2 + \left( \sqrt { \frac { b } { a } } + 1 \right) \log _ { e } \left( \frac { a } { b } \right)$, where a and b are integers with $\operatorname { gcd } ( \mathrm { a } , \mathrm { b } ) = 1$, then $11 \mathrm { a } + 18 \mathrm {~b}$ is equal to $\_\_\_\_$

The value of $\lim_{n \rightarrow \infty}\left(\sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k+3)!}\right)$ is:
(1) $4/3$
(2) $2$
(3) $7/3$
(4) $5/3$