Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$ and $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.
Deduce from the previous questions that for every integer $r \geqslant 2$,
$$S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } } = \frac { ( - 1 ) ^ { r } } { ( r - 1 ) ! } \int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } \frac { \ln ( 1 - t ) } { 1 - t } \mathrm {~d} t$$