We denote $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. We denote $g$ the function defined on $[ - 1 , + \infty [$ by
$$g ( x ) = \sum _ { n = 2 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$$
Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $[ - 1 , + \infty [$.
Specify in particular the value of $g ^ { ( k ) } ( 0 )$ as a function of $\zeta ( k + 1 )$ for every integer $k \geqslant 1$.