Let $\left\{ f _ { n } : \mathbb { R } \longrightarrow \mathbb { R } \right\}$ be a sequence of continuous functions. Let $x _ { n } \longrightarrow x$ be a convergent sequence of reals. If $f _ { n } \longrightarrow f$ uniformly then $f _ { n } \left( x _ { n } \right) \longrightarrow f ( x )$.

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 $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Let $\epsilon \in ] 0 , \pi [$, and $k \in \mathbb { N }$. Prove that for every $h \in C _ { 2 \pi } ( \mathbb { R } ; \mathbb { C } )$ that is of class $C ^ { 1 }$ on $\mathbb { R }$ and every real $u$, we have: $$\int _ { 0 } ^ { 2 \pi } R _ { k } ( u - t ) h ( t ) d t = \int _ { 0 } ^ { 2 \pi } R _ { k } \left( t _ { 1 } \right) h \left( u - t _ { 1 } \right) d t _ { 1 }$$ and $$\left| \int _ { 0 } ^ { 2 \pi } R _ { k } ( u - t ) h ( t ) d t - h ( u ) \right| \leq 2 \left\| h ^ { \prime } \right\| \epsilon + 4 \pi \| h \| d _ { k } ( \epsilon )$$ (We recall that $\int _ { 0 } ^ { 2 \pi } R _ { k } \left( t _ { 1 } \right) d t _ { 1 } = 1$ and that $\| h \|$ is defined at the beginning of the problem statement. To establish the inequality, one may use that $h \left( u - t _ { 1 } \right) = h \left( u - t _ { 1 } + 2 \pi \right)$ when $t _ { 1 } \in [ 2 \pi - \epsilon , 2 \pi ]$).

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. For every real $a \in ]0, \mathrm{e}[$, justify that the sequence of functions $(w_n)$ converges uniformly on $[0, a]$ to the function $W$.

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Does the sequence of functions $(w_n)$ converge uniformly to $W$ on $[0, \mathrm{e}]$?

We recall that the sequence $\left(\left(\sum_{k=1}^{n} k^{-1}\right) - \ln(n)\right)_{n \geqslant 1}$ converges. We denote $\gamma$ its limit. Let $n \in \mathbb{N}^*$. For $x \in ]0, +\infty[$, we set $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the sequence of functions $\left(\Gamma_n\right)_{n \geqslant 1}$ converges pointwise on $]0, +\infty[$ to a function $\Gamma$ from $]0, +\infty[$ to $]0, +\infty[$.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the function $\Gamma$ is of class $\mathscr{C}^2$ and that, for all $x \in ]0, +\infty[$, $$(\ln(\Gamma))''(x) = \sum_{k=0}^{+\infty} \frac{1}{(x+k)^2}.$$

Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. Show that there exists a sequence of polynomials $\left( Q _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ that converges uniformly towards $f$ on $I$ and such that, for all $n \in \mathbb { N } ^ { * }$, the function $Q _ { n }$ does not coincide with $f$ at any point of $I$, except possibly at zero: $$\forall n \in \mathbb { N } ^ { * } , \quad \forall x \in I \backslash \{ 0 \} , \quad Q _ { n } ( x ) \neq \exp ( x ).$$

We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$ For $n \in \mathbf{N}^*$, set $a_n = \frac{1}{n+1}$ and $b_n = \frac{n}{n+1}$.
Establish the pointwise convergence of the sequence of applications $(\Psi_n)_{n \in \mathbf{N}^*}$, from $]0, \pi]$ to $\mathbf{R}$, defined by: $$\forall n \in \mathbf{N}^*, \forall t \in ]0, \pi], \Psi_n(t) = \ln(a_n^2 \cos^2 t + b_n^2 \sin^2 t)$$ Deduce $f''(0)$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $k \in \mathbb{N}$ and $n \geqslant k+2$, we have
$$\left\|\psi_{n+1}^{(k)} - \psi_{n}^{(k)}\right\|_{\infty} \leqslant \frac{\mu_{n+1}}{2} \left\|\psi_{k+1}^{(k+1)}\right\|_{\infty}.$$
Deduce that for all $k \in \mathbb{N}$, the sequence of functions $\psi_{n}^{(k)}$ converges uniformly on $\mathbb{R}$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that the limit $f = \lim_{n \rightarrow \infty} \psi_{n}$ is of class $C^{\infty}$, and that for all $k \geqslant 0$ we have
$$\left\|f^{(k)}\right\|_{\infty} \leqslant \left\|\psi_{k}^{(k)}\right\|_{\infty}.$$

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be defined by $$f(x) = \lim_{n \rightarrow \infty} \cos^{n}\left(\frac{1}{n^{x}}\right)$$
(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.