We fix a natural number $p$ greater than or equal to 2. Show that every upper triangular and invertible matrix admits at least one upper triangular $p$-th root.
One may prove by induction on $n \geqslant 1$ the following property: $$\forall B \in \mathcal { T } _ { n } ( \mathbb { C } ) \cap \mathrm { GL } _ { n } ( \mathbb { C } ) , \quad \exists A \in \mathcal { T } _ { n } ( \mathbb { C } ) \quad \text { such that } \left\{ \begin{array} { l } A ^ { p } = B \\ \forall ( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } , \frac { a _ { i , i } } { a _ { j , j } } \notin \mathcal { V } _ { p } \end{array} \right.$$

Prove by induction that, for all integers $n \in \mathbb{N}^{*}$, there exists a unique polynomial $R_{n} \in \mathbb{R}_{n}[X]$ such that, for all $x \in ]-1,1[$, $$\sum_{p=1}^{+\infty} p^{n} x^{p} = \frac{R_{n}(x)}{(1-x)^{n+1}}$$