Let $f$ be the function defined by
$$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$
Show that there exists $A \in \mathbb{R}_{+}^{*}$ such that
$$\forall k \in \mathbb{N}^{*}, \forall x \in {]-1,1[}, \quad \left|f^{(k)}(x)\right| \leqslant k! \left(A + \frac{1}{(x+1)^{k+1}}\right)$$