We define the sequence $\left(u_{n}\right)$ as follows: for every natural integer $n$, $u_{n} = \int_{0}^{1} \frac{x^{n}}{1+x} \mathrm{~d}x$.

1. Calculate $u_{0} = \int_{0}^{1} \frac{1}{1+x} \mathrm{~d}x$.
2. a) Prove that, for every natural integer $n$, $u_{n+1} + u_{n} = \frac{1}{n+1}$. b) Deduce the exact value of $u_{1}$.
3. a) Copy and complete the algorithm below so that it displays as output the term of rank $n$ of the sequence $(u_{n})$ where $n$ is a natural integer entered as input by the user.
   

   Variables :	$i$ and $n$ are natural integers, $u$ is a real number
   Input :	Enter $n$
   Initialization :	Assign to $u$ the value ...
   Processing :	\begin{tabular}{l} For $i$ varying from 1 to... | Assign to $u$ the value . . .
   End For

   \hline & \hline Output : & Display $u$ \hline \end{tabular}
   b) Using this algorithm, the following table of values was obtained:
   

   $n$	0	1	2	3	4	5	10	50	100
   $u_{n}$	0,6931	0,3069	0,1931	0,1402	0,1098	0,0902	0,0475	0,0099	0,0050

   
   What conjectures concerning the behavior of the sequence $(u_{n})$ can be made?
4. a) Prove that the sequence $(u_{n})$ is decreasing. b) Prove that the sequence $(u_{n})$ is convergent.
5. We call $\ell$ the limit of the sequence $(u_{n})$. Prove that $\ell = 0$.

The purpose of this exercise is to study the sequence $(u_n)$ defined by the value of its first term $u_1$ and, for every natural number $n$ greater than or equal to 1, by the relation: $$u_{n+1} = (n+1)u_n - 1$$
Part A

1. Verify, by detailing the calculation, that if $u_1 = 0$ then $u_4 = -17$.
2. Copy and complete the algorithm below so that by first entering in $U$ a value of $u_1$ it calculates the terms of the sequence $(u_n)$ from $u_2$ to $u_{13}$.
   For $N$ going from 1 to 12 $$U \leftarrow$$ End For
   
3. This algorithm was executed for $u_1 = 0.7$ then for $u_1 = 0.8$.
   Here are the values obtained.
   

   For $u_1 = 0.7$	For $u_1 = 0.8$
   0.4	0.6
   0.2	0.8
   -0.2	2.2
   -2	10
   -13	59
   -92	412
   -737	3295
   -6634	29654
   -66341	296539
   -729752	3261928
   -8757025	39143135
   -113841326	508860754

   
   What appears to be the limit of this sequence if $u_1 = 0.7$? And if $u_1 = 0.8$?

Part B
We consider the sequence $(I_n)$ defined for every natural number $n$, greater than or equal to 1, by: $$I_n = \int_0^1 x^n \mathrm{e}^{1-x} \mathrm{~d}x$$ We recall that the number e is the value of the exponential function at 1, that is to say that $\mathrm{e} = \mathrm{e}^1$.

1. Prove that the function $F$ defined on the interval $[0;1]$ by $F(x) = (-1-x)\mathrm{e}^{1-x}$ is an antiderivative on the interval $[0;1]$ of the function $f$ defined on the interval $[0;1]$ by $f(x) = x\mathrm{e}^{1-x}$.
2. Deduce that $I_1 = \mathrm{e} - 2$.
3. It is admitted that, for every natural number $n$ greater than or equal to 1, we have: $$I_{n+1} = (n+1)I_n - 1.$$ Use this formula to calculate $I_2$.
4. a. Justify that, for every real number $x$ in the interval $[0;1]$ and for every natural number $n$ greater than or equal to 1, we have: $0 \leqslant x^n \mathrm{e}^{1-x} \leqslant x^n \mathrm{e}$. b. Justify that: $\int_0^1 x^n \mathrm{e} \, \mathrm{d}x = \dfrac{\mathrm{e}}{n+1}$. c. Deduce that, for every natural number $n$ greater than or equal to 1, we have: $0 \leqslant I_n \leqslant \dfrac{\mathrm{e}}{n+1}$. d. Determine $\lim_{n \rightarrow +\infty} I_n$.

Part C
In this part, we denote by $n!$ the number defined, for every natural number $n$ greater than or equal to 1, by: $1! = 1$, $2! = 2 \times 1$, and if $n \geqslant 3$: $n! = n \times (n-1) \times \ldots \times 1$. And, more generally: $(n+1)! = (n+1) \times n!$

1. Prove by induction that, for every natural number $n$ greater than or equal to 1, we have: $$u_n = n! \left(u_1 - \mathrm{e} + 2\right) + I_n$$ We recall that, for every natural number $n$ greater than or equal to 1, we have: $$u_{n+1} = (n+1)u_n - 1 \quad \text{and} \quad I_{n+1} = (n+1)I_n - 1.$$
2. It is admitted that: $\lim_{n \rightarrow +\infty} n! = +\infty$. a. Determine the limit of the sequence $(u_n)$ when $u_1 = 0.7$. b. Determine the limit of the sequence $(u_n)$ when $u_1 = 0.8$.

We consider the sequence $\left( I _ { n } \right)$ defined by $I _ { 0 } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 - x } \mathrm {~d} x$ and for every non-zero natural number $n$
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { x ^ { n } } { 1 - x } \mathrm {~d} x$$

1. Show that $I _ { 0 } = \ln ( 2 )$.
2. a. Calculate $I _ { 0 } - I _ { 1 }$. b. Deduce $I _ { 1 }$.
3. a. Show that, for every natural number $n , I _ { n } - I _ { n + 1 } = \frac { \left( \frac { 1 } { 2 } \right) ^ { n + 1 } } { n + 1 }$. b. Propose an algorithm to determine, for a given natural number $n$, the value of $I _ { n }$.
4. Let $n$ be a non-zero natural number.

It is admitted that if $x$ belongs to the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$ then $0 \leqslant \frac { x ^ { n } } { 1 - x } \leqslant \frac { 1 } { 2 ^ { n - 1 } }$. a. Show that for every non-zero natural number $n$, $0 \leqslant I _ { n } \leqslant \frac { 1 } { 2 ^ { n } }$. b. Deduce the limit of the sequence ( $I _ { n }$ ) as $n$ tends to $+ \infty$.
5. For every non-zero natural number $n$, we set
$$S _ { n } = \frac { 1 } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 } } { 2 } + \frac { \left( \frac { 1 } { 2 } \right) ^ { 3 } } { 3 } + \ldots + \frac { \left( \frac { 1 } { 2 } \right) ^ { n } } { n }$$
a. Show that for every non-zero natural number $n$, $S _ { n } = I _ { 0 } - I _ { n }$. b. Determine the limit of $S _ { n }$ as $n$ tends to $+ \infty$.

Part 1
We consider the function $f$ defined on the set of real numbers $\mathbb{R}$ by: $$f(x) = \left(x^2 - 4\right)\mathrm{e}^{-x}$$ We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function.

1. Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
2. Justify that for all real $x$, $f'(x) = \left(-x^2 + 2x + 4\right)\mathrm{e}^{-x}$.
3. Deduce the variations of the function $f$ on $\mathbb{R}$.

Part 2
We consider the sequence $(I_n)$ defined for all natural integer $n$ by $I_n = \int_{-2}^{0} x^n \mathrm{e}^{-x}\,\mathrm{d}x$.

1. Justify that $I_0 = \mathrm{e}^2 - 1$.
2. Using integration by parts, demonstrate the equality: $$I_{n+1} = (-2)^{n+1}\mathrm{e}^2 + (n+1)I_n$$
3. Deduce the exact values of $I_1$ and $I_2$.

Part 3

1. Determine the sign on $\mathbb{R}$ of the function $f$ defined in Part 1.
2. The curve $\mathscr{C}_f$ of the function $f$ is represented in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. The domain $D$ of the shaded region is bounded by the curve $\mathscr{C}_f$, the $x$-axis and the $y$-axis. Calculate the exact value, in square units, of the area $S$ of the domain $D$.

For every natural number $n$, we consider the following integrals:
$$I_n = \int_0^{\pi} \mathrm{e}^{-nx} \sin(x) \mathrm{d}x, \quad J_n = \int_0^{\pi} \mathrm{e}^{-nx} \cos(x) \mathrm{d}x$$

1. Calculate $I_0$.
2. a. Justify that, for every natural number $n$, we have $I_n \geqslant 0$. b. Show that, for every natural number $n$, we have $I_{n+1} - I_n \leqslant 0$. c. Deduce from the two previous questions that the sequence $(I_n)$ converges.
3. a. Show that, for every natural number $n$, we have: $$I_n \leqslant \int_0^{\pi} \mathrm{e}^{-nx} \mathrm{d}x$$ b. Show that, for every natural number $n \geqslant 1$, we have: $$\int_0^{\pi} \mathrm{e}^{-nx} \mathrm{d}x = \frac{1 - \mathrm{e}^{-n\pi}}{n}.$$ c. Deduce from the two previous questions the limit of the sequence $(I_n)$.
4. a. By integrating by parts the integral $I_n$ in two different ways, establish the two following relations, for every natural number $n \geqslant 1$: $$I_n = 1 + \mathrm{e}^{-n\pi} - nJ_n \quad \text{and} \quad I_n = \frac{1}{n}J_n$$ b. Deduce that, for every natural number $n \geqslant 1$, we have $$I_n = \frac{1 + \mathrm{e}^{-n\pi}}{n^2 + 1}$$
5. It is desired to obtain the rank $n$ from which the sequence $(I_n)$ becomes less than $0.1$. Copy and complete the fifth line of the Python script below with the appropriate command. \begin{verbatim} from math import * def seuil() : n = 0 I = 2 ... n=n+1 I =(1+exp(-n*pi))/(n*n+1) return n \end{verbatim}

Evaluate $\int_{0}^{\infty} \left(1 + x^{2}\right)^{-(m+1)} dx$, where $m$ is a natural number.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$. For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ and $T_0(x) = 1$. The space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $(f|g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
Calculate $(T_m | T_n)$ for all $(m, n) \in \mathbb{N} \times \mathbb{N}$. What can we deduce from this?

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

The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Justify that the function $\Gamma$ is of class $\mathcal{C}^{1}$ and strictly positive on $]0, +\infty[$.

The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$.

The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Calculate $\Gamma(n)$ for all natural integers $n$, $n \geqslant 1$.

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 define for all real $x > 0$ the sequence $(I_{n}(x))_{n \geqslant 1}$ by: $$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt$$ Show that, for all $x > 0$, $$\lim_{n \rightarrow +\infty} I_{n}(x) = \Gamma(x)$$

We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by: $$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Show that, for all integers $n$, $n \geqslant 0$, $$\forall x > 0, \quad J_{n+1}(x) = \frac{n+1}{x} J_{n}(x+1)$$

We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by: $$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Deduce that, for all $x > 0$, $$J_{n}(x) = \frac{n!}{x(x+1) \cdots (x+n-1)(x+n)}$$

We define for all real $x > 0$ the sequences $(I_{n}(x))_{n \geqslant 1}$ and $(J_{n}(x))_{n \geqslant 0}$ by: $$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt, \qquad J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Establish Euler's identity: $$\forall x > 0, \quad \Gamma(x) = \lim_{n \rightarrow +\infty} \frac{n! \, n^{x}}{x(x+1) \cdots (x+n)}$$

Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Calculate $\int _ { 0 } ^ { \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } \sin t \, d t$. Deduce that $c _ { k } \leq \frac { k + 1 } { 4 }$.

Deduce that $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) d \theta = \begin{cases} 4 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$

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 $(2k)!$).

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in A$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}$.
b) Let $m \in \mathbb{R}$. Show that $T_{m}$ is continuous on $\mathbb{R}_{+}^{*}$.
c) Show that for $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$, $$T_{m}(x) \geq e^{-1} \int_{0}^{1} t^{m} e^{-x/t} dt$$ Deduce the value of $\lim_{x \rightarrow 0^{+}} T_{m}(x)$ when $m \notin A$, using the change of variable $w = x/t$.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in \mathbb{R}$. Show that $T_{m}$ is of class $C^{1}$ on $\mathbb{R}_{+}^{*}$, and calculate $T_{m}^{\prime}$ in terms of $T_{m-1}$.
b) Let $m \in \mathbb{R}$. Is the function $T_{m}$ of class $C^{\infty}$ on $\mathbb{R}_{+}^{*}$? What is the monotonicity of $T_{m}$ on $\mathbb{R}_{+}^{*}$? Is the function $T_{m}$ convex on $\mathbb{R}_{+}^{*}$?
c) Discuss as a function of $m \in \mathbb{R}$ the right-differentiability of $T_{m}$ at 0.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$. Calculate $T_{m}(x)$ in terms of $T_{m-2}(x)$ and $T_{m-3}(x)$. For this, you may consider the quantity $$\int_{A}^{B} t^{m-1}\left(2t - x/t^{2}\right) e^{-\left(t^{2}+x/t\right)} dt$$ for $0 < A < B$.
b) Let $m \in \mathbb{R}$. Find a relation between $x T_{m}^{\prime\prime\prime}(x)$, $T_{m}^{\prime\prime}(x)$ and $T_{m}(x)$ for $x \in \mathbb{R}_{+}^{*}$.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$. Perform the change of variable $t = 1/u$ in the integral defining $T_{m}$. Justify the calculation carefully.
b) Let $n \in \mathbb{N}$. Justify the existence of the quantity $\int_{0}^{\infty} u^{n} e^{-u} du$ and calculate it.
c) Show that for $m \in \mathbb{N} - \{0,1\}$, $T_{-m}(1) \leq (m-2)!$.
d) Let $k \in \mathbb{N}$. Show that the radius of convergence $R$ of the power series $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$ satisfies $R \geq 1$.
e) Let $k \in \mathbb{N}$. Show that for $x \in ]-1,1[$, $$T_{k}(1+x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$$

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ For $t, x \in \mathbb{R}_{+}^{*}$, we set $g(t,x) = t^{2} + x/t$.
a) Show that for $x \in \mathbb{R}_{+}^{*}$, the function $t \in \mathbb{R}_{+}^{*} \mapsto g(t,x)$ is convex and admits a unique minimum at $t = M(x)$, which we will determine. Calculate $g(M(x),x)$.
b) Let $m \in \mathbb{R}_{+}$, and $x \in \mathbb{R}_{+}^{*}$. Show using the inequality $M(x) \leq x^{1/3}$ that $$T_{m}(x) \leq \int_{0}^{x^{1/3}} t^{m} e^{-g(t,x)} dt + e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right)$$
c) Show that for every $\varepsilon > 0$ (and $m \in \mathbb{R}_{+}$), we can find $C > 0$ such that $$\forall t \geq 1, \quad t^{m} \leq C t e^{\varepsilon t^{2}}$$ Deduce that for every $\varepsilon > 0$, $$e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right) = O_{x \rightarrow \infty}\left(e^{-\left[\frac{39}{20} - \varepsilon\right] x^{2/3}}\right)$$
d) Show that $$T_{m}(x) = O_{x \rightarrow \infty}\left(x^{\frac{m+1}{3}} e^{-3(x/2)^{2/3}}\right)$$

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Show that for $x \in \mathbb{R}_{+}^{*}$, $$T_{-1}(x) \leq \int_{0}^{1} e^{-1/u^{2}} \frac{du}{u} + \int_{1}^{\infty} e^{-xu} \frac{du}{u}$$ Deduce that $T_{-1}(x) \leq 2$ for $x \geq 1$ and that $$T_{-1}(x) \leq 2 + \int_{x}^{1} e^{-w} \frac{dw}{w} \leq 2 - \ln x$$ if $0 < x \leq 1$.
b) Let $L \in [0,1]$, and $\rho \in C([0,L])$. We set $$[F(\rho)](x) = \int_{0}^{L} \rho(y) T_{-1}(|x-y|) dy$$ Show that $[F(\rho)](x)$ is well-defined for $x \in [0,L]$ and that $$\|F(\rho)\|_{\infty} \leq (4L + 2L|\ln L|) \|\rho\|_{\infty}.$$