Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$, $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges, and $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. Deduce that $S _ { 2 } = \frac { 1 } { 2 } \int _ { 0 } ^ { 1 } \frac { ( \ln t ) ^ { 2 } } { 1 - t } \mathrm {~d} t$ then find the value of $S _ { 2 }$ as a function of $\zeta ( 3 )$.
Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$, $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges, and $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$.
Deduce that $S _ { 2 } = \frac { 1 } { 2 } \int _ { 0 } ^ { 1 } \frac { ( \ln t ) ^ { 2 } } { 1 - t } \mathrm {~d} t$ then find the value of $S _ { 2 }$ as a function of $\zeta ( 3 )$.