We denote $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$. We have shown that for every real $x > -1$, $\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$.
Prove that for every $x$ in $] - 1,1 [$,
$$\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } ( - 1 ) ^ { n + 1 } \zeta ( n + 1 ) x ^ { n }$$