We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, satisfying $\psi ( x + 1 ) - \psi ( x ) = \frac{1}{x}$ for all $x > 0$.
Deduce that, for every real $x > - 1$,
$$\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$$