Show that the function $t \rightarrow e^{-t} t^{x-1}$ is integrable on $]0, +\infty[$ if, and only if, $x > 0$.

We denote by $(f_{n})_{n \geqslant 1}$ the sequence of functions defined on $]0, +\infty[$ by: $$f_{n}(t) = \begin{cases} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} & \text{if } t \in ]0, n[ \\ 0 & \text{if } t \geqslant n \end{cases}$$ Show that for all integers $n$, $n \geqslant 1$, the function $f_{n}$ is continuous and integrable on $]0, +\infty[$.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ We may freely use the formula $T_{0}(0) = \frac{\sqrt{\pi}}{2}$.
a) Show that if $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, the integral defining $T_{m}(x)$ is convergent.
b) What is the interval $A$ of $m \in \mathbb{R}$ such that the integral defining $T_{m}(0)$ is convergent?
c) Calculate $T_{2k}(0)$ and $T_{2k+1}(0)$ for $k \in \mathbb{N}$ (in terms of $k!$ and $(2k)!$).

What is the domain of definition $\mathcal{D}$ of the function $\Gamma$, where for $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$?

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Justify the existence of the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ and specify the monotonicity of the subsequence $\left(u_{2n}\right)_{n \in \mathbb{N}^{*}}$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $\left(J_{n}\right)_{n \in \mathbb{N}^{*}}$ is a well-defined sequence and that it is increasing and convergent.
We will set $a_{k} = \frac{1}{2k-1}$ and express the expectation of $\left|T_{n}\right|$ using the method of question II.A.4.

Show that for all natural integer $p$, the integral $$I _ { p } = \int _ { - \infty } ^ { + \infty } e ^ { - ( t - p \pi ) ^ { 2 } } \sin t \mathrm {~d} t$$ is absolutely convergent and that it equals zero.

Consider the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$, and let $\tilde{h}$ denote its restriction to $\left]0, \frac{1}{2}\right]$. Show that the function $h$ is integrable on $]0,1[$ and that: $$\int_0^1 h(t)\, dt = 2\int_0^{\frac{1}{2}} \tilde{h}(t)\, dt.$$

Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$ and $S \in \mathbb { R }$. Prove that $$\left( \lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = S \text { and } f ( t ) \underset { t \rightarrow + \infty } { = } O \left( \frac { 1 } { t } \right) \right) \Rightarrow \left( \int _ { 0 } ^ { + \infty } f ( x ) d x \text { converges and } \int _ { 0 } ^ { + \infty } f ( x ) d x = S \right)$$
For this, using the notations of question 5 of section 2, one can prove that there exist $M > 0$ and $A > 0$ such that for all $t > 0$ $$\begin{aligned} \left| \int _ { A } ^ { + \infty } f ( x ) g \left( e ^ { - t x } \right) d x - \int _ { A } ^ { + \infty } f ( x ) P _ { 1 } \left( e ^ { - t x } \right) d x \right| & \leqslant M \int _ { A } ^ { + \infty } Q \left( e ^ { - t x } \right) e ^ { - t x } \frac { 1 - e ^ { - t x } } { x } d x \\ & \leqslant M \int _ { 0 } ^ { 1 } Q ( u ) d u \end{aligned}$$