Which of the following series converge?
I. $\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } } { n ! }$
II. $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { n ^ { 100 } }$
III. $\sum _ { n = 1 } ^ { \infty } \frac { n + 1 } { ( n ) ( n + 2 ) ( n + 3 ) }$
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

For what values of $p$ will both series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 p } }$ and $\sum _ { n = 1 } ^ { \infty } \left( \frac { p } { 2 } \right) ^ { n }$ converge?
(A) $- 2 < p < 2$ only
(B) $- \frac { 1 } { 2 } < p < \frac { 1 } { 2 }$ only
(C) $\frac { 1 } { 2 } < p < 2$ only
(D) $p < \frac { 1 } { 2 }$ and $p > 2$
(E) There are no such values of $p$.

The series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ where $a _ { n } = ( - 1 ) ^ { n + 1 } n ^ { 4 } e ^ { - n ^ { 2 } }$
(a) has unbounded partial sums;
(b) is absolutely convergent;
(c) is convergent but not absolutely convergent;
(d) is not convergent, but partial sums oscillate between $-1$ and $+1$.

Which of the following is/are true for a series of real numbers $\sum a_{n}$?
(A) If $\sum a_{n}$ converges then $\sum a_{n}^{2}$ converges;
(B) If $\sum a_{n}^{2}$ converges then $\sum a_{n}$ converges;
(C) if $\sum a_{n}^{2}$ converges then $\sum \frac{1}{n} a_{n}$ converges;
(D) If $\sum |a_{n}|$ converges then $\sum \frac{1}{n} a_{n}$ converges;

Consider the improper integral $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \log x ) ^ { 2 } } d x$ and the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { 1 } { k ( \log k ) ^ { 2 } }$. Which of the following is/are true?
(A) The integral converges but the series does not converge.
(B) The integral does not converge but the series converges.
(C) Both the integral and the series converge.
(D) The integral and the series both fail to converge.

Let $a _ { n } , n \geq 1$, be a sequence of positive real numbers such that $a _ { n } \longrightarrow \infty$ as $n \longrightarrow \infty$. Then which of the following are true?
(A) There exists a natural number $M$ such that
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( a _ { n } \right) ^ { M } } \in \mathbb { R }$$
(B)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n ^ { 2 } a _ { n } \right) } \in \mathbb { R } .$$
(C)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n a _ { n } \right) } \in \mathbb { R }$$
(D) For all positive real numbers $R$,
$$\sum _ { n = 1 } ^ { \infty } \frac { R ^ { n } } { \left( a _ { n } \right) ^ { n } } \in \mathbb { R } .$$

For a sequence $a _ { i }$ of real numbers, we say that $\sum a _ { i }$ converges if $\lim _ { n \rightarrow \infty } \left( \sum _ { i = 1 } ^ { n } a _ { i } \right)$ is finite. In this question all $a _ { i } > 0$.
Statements
(21) If $\sum a _ { i }$ converges, then $a _ { i } \rightarrow 0$ as $i \rightarrow \infty$. (22) If $a _ { i } < \frac { 1 } { i }$ for all $i$, then $\sum a _ { i }$ converges. (23) If $\sum a _ { i }$ converges, then $\sum ( - 1 ) ^ { i } a _ { i }$ also converges. (24) If $\sum a _ { i }$ does not converge, then $\sum i \tan \left( a _ { i } \right)$ cannot converge.

Let $\mathbb { R } ^ { + } = \{ x \in \mathbb { R } : x \geq 0 \}$. For $x \in \mathbb { R } ^ { + }$, denote by $\operatorname{FRAC}( x )$ the fractional part of $x$, i.e., $x - n$ where $n$ is the largest integer that is less than or equal to $x$. Consider the series $\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { FRAC } ( x / n ) } { n }$. Pick the correct statement(s) from below.
(A) The above series converges for all $x \in \mathbb { R } ^ { + } - \mathbb { Z }$.
(B) The above series diverges for some non-negative integer $x$.
(C) The above series defines a continuous function in a neighbourhood of $\frac { 1 } { 2 }$.
(D) The above series defines a continuous function in a neighbourhood of 1.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$.
Show that the numerical series $\sum_{n \in \mathbb{N}} n^j \alpha_n$ is convergent.

Deduce the nature of the Riemann series $\sum _ { n \geqslant 1 } \frac { 1 } { n ^ { \alpha } }$ according to the value of $\alpha \in \mathbb { R }$.

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 $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.

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 $\mathbb{R}_{+}$?

Does there exist a function $g \in C_{b}(\mathbb{R})$ such that for every function $f$ in $L^{1}(\mathbb{R})$, we have $f * g = f$?

Show that there exist two functions $f$ and $g$ in $L^{1}(\mathbb{R})$ such that $f * g(0)$ is not defined.

We define $$\varphi(x) = \begin{cases} \left(\frac{\sin x}{x}\right)^{2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Justify that $\varphi \in L^{1}(\mathbb{R})$.

What inclusion exists between the sets $E$ and $E^{\prime}$, where $E$ is the set of real numbers $x$ for which the application $t \mapsto f(t)e^{-\lambda(t)x}$ is integrable on $\mathbb{R}^+$, and $E^{\prime}$ is the set of real numbers $x$ for which the integral $\int_0^{+\infty} f(t)e^{-\lambda(t)x}\,dt$ converges?

Compare $E$ and $E^{\prime}$ in the case where $f$ is positive.

In the three following cases, determine $E$.
II.B.1) $f(t) = \lambda^{\prime}(t)$, with $\lambda$ assumed to be of class $C^1$.
II.B.2) $f(t) = e^{t\lambda(t)}$.
II.B.3) $f(t) = \dfrac{e^{-t\lambda(t)}}{1+t^2}$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Determine $E$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $f$ admits a finite limit $l$ at $+\infty$.
IV.D.1) Show that $E$ contains $\mathbb{R}^{+*}$.
IV.D.2) Show that $xLf(x)$ tends to $l$ at $0^+$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E$ does not contain 0.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E = ]0, +\infty[$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E^{\prime}$ contains 0.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients and we use the transformation $L$ applied to elements of $\mathcal{P}$ for the study of an operator $U$.
Let $P$ and $Q$ be two elements of $\mathcal{P}$.
Show that the integral $\displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, where $\bar{P}$ is the polynomial whose coefficients are the conjugates of those of $P$, converges.

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.