In this question, $\tau$ is an element of $\mathbb{R}^{+*}$ and $X$ satisfies $(C_{\tau})$.
a) Show that the set of real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$ is an interval $I$ containing $[-\tau, \tau]$. For $t$ in $I$, we denote $\varphi_{X}(t) = E\left(\mathrm{e}^{tX}\right)$.
b) Show that if $X(\Omega)$ is finite, $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
c) Suppose now that $X(\Omega)$ is a countably infinite set. We denote $X(\Omega) = \left\{x_{n} ; n \in \mathbb{N}^{*}\right\}$ where $\left(x_{n}\right)_{n \in \mathbb{N}^{*}}$ is a sequence of pairwise distinct real numbers and we set for all $n \in \mathbb{N}^{*}$, $p_{n} = P\left(X = x_{n}\right)$. Using the results established in question I.A.3 and two theorems relating to series of functions which you will state completely, show that $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
d) Verify that for $t$ in the interior of $I$ and $k$ in $\mathbb{N}$, $\varphi_{X}^{(k)}(t) = E\left(X^{k} \mathrm{e}^{tX}\right)$.
e) Let $\psi_{X} = \frac{\varphi_{X}^{\prime}}{\varphi_{X}}$. Show that $\psi_{X}$ is increasing on $I$ and that, if $X$ is not almost surely equal to a constant, $\psi_{X}$ is strictly increasing on $I$.