Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$. We denote $\zeta$ the function defined for $x > 1$ by $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$. Let $r$ be a natural integer. For which values of $r$ is the series $\sum _ { n \geqslant 1 } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ convergent? In the rest of the problem we will denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.
Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$. We denote $\zeta$ the function defined for $x > 1$ by $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$.
Let $r$ be a natural integer. For which values of $r$ is the series $\sum _ { n \geqslant 1 } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ convergent?
In the rest of the problem we will denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.