We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set
$$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$
We also introduce the function
$$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$
Show, for all real $t > 0$, the existence of $S _ { n , \alpha } ( t )$, its strict positivity, and the identity
$$\int _ { 0 } ^ { + \infty } \varphi _ { n , \alpha } ( x ) \mathrm { d } x = t ^ { n + 1 } S _ { n , \alpha } ( t ) - \sum _ { k = 0 } ^ { + \infty } \int _ { k t } ^ { ( k + 1 ) t } ( x - k t ) \varphi _ { n , \alpha } ^ { \prime } ( x ) \mathrm { d } x$$
Deduce that
$$S _ { n , \alpha } ( t ) = \frac { 1 } { t ^ { n + 1 } } \int _ { 0 } ^ { + \infty } \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } } \mathrm {~d} x + O \left( \frac { 1 } { t ^ { n } } \right) \quad \text { as } t \rightarrow 0 ^ { + }$$