Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$ Let $Q(X)$ be the reciprocal polynomial of $P(X)$ defined by $Q(X) = X^d P\left(\frac{1}{X}\right)$.
We define the function $f : \mathbb { R } \backslash \left( \mathbb { R } \cap \left\{ \frac { 1 } { \lambda _ { 1 } } , \ldots , \frac { 1 } { \lambda _ { d } } \right\} \right) \rightarrow \mathbb { C }$ by $f ( x ) = \frac { Q ^ { \prime } ( x ) } { Q ( x ) }$.
Show that there exists $r > 0$ such that $f$ is expandable as a power series on $] - r , r [$, and that the power series expansion of $f$ at 0 is written as: $$\forall x \in ] - r , r \left[ , \quad f ( x ) = - \sum _ { n = 0 } ^ { \infty } N _ { n + 1 } x ^ { n } \right.$$