Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$.
Deduce that the function
$$t \mapsto - \frac { \ln ( 1 - t ) } { 1 - t }$$
is expandable as a power series on $] - 1,1 [$ and specify its power series expansion using the real numbers $H _ { n }$.