For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show $$\forall s \in \mathbb{R}, \quad |s| = \frac{2}{\pi} \int_{0}^{\infty} \frac{1 - \cos(st)}{t^{2}} \mathrm{~d}t$$

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 study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $u_{1} = u_{2} = \frac{\pi}{2}$.

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \frac{\sqrt{n}}{2\sqrt{2}} v_{n} \quad \text{with} \quad v_{n} = \int_{0}^{\infty} \frac{1 - (\cos(\sqrt{2u/n}))^{n}}{u\sqrt{u}} \mathrm{~d}u$$

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $$\left. \left. \forall (n, u) \in \mathbb{N}^{*} \times \right] 0, 1 \right], \quad \left| 1 - (\cos(\sqrt{2u/n}))^{n} \right| \leqslant u$$

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ and $v_{n} = \int_{0}^{\infty} \frac{1 - (\cos(\sqrt{2u/n}))^{n}}{u\sqrt{u}} \mathrm{~d}u$.
Show that the sequence $\left(v_{n}\right)_{n \in \mathbb{N}^{*}}$ admits a finite limit $l$ satisfying $$l = \int_{0}^{\infty} \frac{1 - \mathrm{e}^{-u}}{u\sqrt{u}} \mathrm{~d}u$$

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ We admit the relation $\int_{0}^{\infty} \frac{\mathrm{e}^{-u}}{\sqrt{u}} \mathrm{~d}u = \sqrt{\pi}$.
Conclude that $u_{n} \sim \sqrt{\frac{n\pi}{2}}$.

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.

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 $J_{n} = \frac{\pi}{2}$ for $1 \leqslant n \leqslant 7$ and that $\left(J_{n}\right)_{n \geqslant 7}$ is strictly increasing.

For $n \in \mathbb{N}$, we set $$I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx.$$ Determine by induction $I_{n}$ for all $n \in \mathbb{N}$.

For $n \in \mathbb{N}$, we set $I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx$.
Show that for $n \geqslant 1$, we have $$I_{n} = \left(\frac{n}{e}\right)^{n} \sqrt{n} \int_{-\sqrt{n}}^{+\infty} \left(1 + \frac{x}{\sqrt{n}}\right)^{n} e^{-x\sqrt{n}}\, dx.$$

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
We denote $$g_{j} = \int_{-1}^{1} P_{j}(x)^{2}\,dx, \quad I_{j} = \int_{-1}^{1} \left(1 - x^{2}\right)^{j}\,dx$$
(a) Establish a relation between $g_{j}$ and $I_{j}$.
(b) Find a relation between $I_{j}$ and $I_{j-1} - I_{j}$, and deduce a recurrence relation for the sequence $\left(I_{j}\right)_{j \in \mathbb{N}}$.
(c) Deduce the value of $I_{j}$, then that of $g_{j}$.

We denote $M_{p} = \int_{-\infty}^{+\infty} x^{2p} \exp\left(-x^{2}\right) \mathrm{d}x$. For $p$ a natural integer, give a relation between $M_{p+1}$ and $M_{p}$ and deduce that, for all $p \in \mathbb{N}$, $$M_{p} = \frac{\sqrt{\pi}(2p)!}{2^{2p} p!}$$

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.

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Show $$\forall n \in \llbracket 2, +\infty\llbracket, \forall x \in \mathbb{R}, \quad \left(1 - \frac{4x^2}{n^2}\right) I_n(x) = \frac{n-1}{n} I_{n-2}(x) \quad \text{and} \quad \left(1 - \frac{4x^2}{n^2}\right) \frac{I_n(x)}{I_n(0)} = \frac{I_{n-2}(x)}{I_{n-2}(0)}.$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Show $$\forall n \in \mathbb{N}^{\star}, \forall x \in \mathbb{R}, \quad \sin(\pi x) = \pi x \frac{I_{2n}(x)}{I_{2n}(0)} \prod_{k=1}^{n} \left(1 - \frac{x^2}{k^2}\right)$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Using the result of Q20, deduce $$\forall n \in \mathbb{N}^{\star}, \forall x \in ]0,1[, \quad \cos(\pi x) = \frac{1}{2} \frac{I_{4n}(2x)}{I_{4n}(0)} \frac{I_{2n}(0)}{I_{2n}(x)} \prod_{p=1}^{n} \left(1 - \frac{4x^2}{(2p-1)^2}\right)$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$, and for $x \in J = [0,1/2[$, $$S_n(x) = \sum_{p=1}^{+\infty} \left(\sum_{k=n+1}^{+\infty} \frac{2^{2p+1} x^{2p-1}}{(2k-1)^{2p}}\right).$$ Show $$\forall n \in \mathbb{N}^{\star}, \forall x \in J, \quad \pi \tan(\pi x) + S_n(x) = -\frac{2I_{4n}^{\prime}(2x)}{I_{4n}(2x)} + \frac{I_{2n}^{\prime}(x)}{I_{2n}(x)} + \sum_{p=1}^{+\infty} 2\left(2^{2p} - 1\right) \zeta(2p) x^{2p-1}.$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Using the inequality $t\cos(t) \leqslant \sin(t)$ for $t \in [0, \pi/2]$, deduce $$\forall n \in \mathbb{N}^{\star}, \forall x \in [0,1], \quad 0 \leqslant -I_n^{\prime}(x) \leqslant \frac{4x}{n} I_n(x)$$ then, for $x \in [0,1]$, the limit $\lim_{n \rightarrow +\infty} \frac{I_n^{\prime}(x)}{I_n(x)}$.

Using the results of Q26 and Q28, deduce the equality $$\forall x \in J, \quad \pi \tan(\pi x) = \sum_{p=1}^{+\infty} 2\left(2^{2p} - 1\right) \zeta(2p) x^{2p-1}$$

We admit the identity $\int_{-\infty}^{+\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}$.
(a) Show that for all integer $n \in \mathbb{N}$, we have $$n! = \int_0^{+\infty} e^{-t} t^n \mathrm{d}t$$
(b) Using the preceding results, recover Stirling's formula giving an asymptotic equivalent of $n!$.

We admit the identity $$\int _ { - \infty } ^ { + \infty } \exp \left( - x ^ { 2 } \right) \mathrm { d } x = \sqrt { \pi }$$
(a) Show that for all integer $n \in \mathbb { N }$, we have $$n ! = \int _ { 0 } ^ { + \infty } e ^ { - t } t ^ { n } \mathrm {~d} t$$
(b) Using the preceding results, recover Stirling's formula giving an asymptotic equivalent of $n !$.

We admit the identities: $$\lim_{a \rightarrow +\infty} \int_0^a \sin(x^2) \mathrm{d}x = \lim_{a \rightarrow +\infty} \int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4}$$
Show that there exist real numbers $c, c' \in \mathbb{R}$ such that, as $a \rightarrow +\infty$, $$\int_0^a \sin(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4} + \frac{c}{a} \cos(a^2) + \frac{c'}{a^3} \sin(a^2) + O\left(\frac{1}{a^5}\right)$$

We admit the identities: $$\lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x = \lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \cos \left( x ^ { 2 } \right) \mathrm { d } x = \frac { \sqrt { 2 \pi } } { 4 }$$
Show that there exist real numbers $c , c ^ { \prime } \in \mathbb { R }$ such that, as $a \rightarrow + \infty$, we have $$\int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x = \frac { \sqrt { 2 \pi } } { 4 } + \frac { c } { a } \cos \left( a ^ { 2 } \right) + \frac { c ^ { \prime } } { a ^ { 3 } } \sin \left( a ^ { 2 } \right) + O \left( \frac { 1 } { a ^ { 5 } } \right) .$$

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$.
We admit that there exist real numbers $d, d' \in \mathbb{R}$ such that, as $a \rightarrow +\infty$, $$\int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4} + \frac{d}{a} \sin(a^2) + \frac{d'}{a^3} \cos(a^2) + O\left(\frac{1}{a^5}\right)$$
Show that, as $t \rightarrow +\infty$, $$\int_{x_0}^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \int_{x_0}^1 \sin(tf(x)) \mathrm{d}x + O\left(\frac{1}{t}\right)$$