For $k \in \mathbf { N } ^ { * }$ and $t \in \mathbf { R } _ { + }$, we set
$$u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { t q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u \quad \text { if } t > 0 , \quad \text { and } \quad u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { q ( u ) } { u } \mathrm {~d} u \quad \text { if } t = 0$$
where $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$.
Let $t \in \mathbf { R } _ { + } ^ { * }$. Show successively that $\left| u _ { k } ( t ) \right| = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { t | q ( u ) | } { e ^ { t u } - 1 } \mathrm {~d} u$, then $u _ { k } ( t ) = ( - 1 ) ^ { k } \left| u _ { k } ( t ) \right|$ for all integer $k \geq 1$, and finally establish that
$$\forall n \in \mathbf { N } ^ { * } , \left| \sum _ { k = n } ^ { + \infty } u _ { k } ( t ) \right| \leq \frac { 1 } { 2 n } .$$
We admit in what follows that this bound also holds for $t = 0$.