We set, for all $n \geq 0$ and all $x \in \mathbf{R}$, $P_n(x) = \sum_{k=0}^{n} \frac{x^k}{k!}$ where $k!$ denotes the factorial of $k$.
Let $A \in \operatorname{Sym}^+(p)$.
(a) Show that for all $(i,j) \in \llbracket 1,p \rrbracket^2$, we have
$$\lim_{n \rightarrow +\infty} P_n[A]_{ij} = \exp\left(A_{ij}\right)$$
(b) Show that $\exp[A] \in \operatorname{Sym}^+(p)$.
(c) Let $u \in \mathbf{R}^p$. Show that $\exp[A] \odot \left(uu^T\right) \in \operatorname{Sym}^+(p)$.