Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$, $r \in ]0, R[$, and $f(x) = \sum_{k=0}^{+\infty} c_k x^k$ for $x \in ]-R, R[$. Let $C \in \mathbb{R}$ be such that $|c_k| \leq C/r^k$ for all $k \in \mathbb{N}$. Deduce that for all $x \in ] - r , r [$ and for all $n \in \mathbb { N }$,
$$\left| f ^ { ( n ) } ( x ) \right| \leqslant \frac { n ! r C } { ( r - | x | ) ^ { n + 1 } }.$$