For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set $$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$ Let $t \in \mathbf{R}_+$. Show successively that $|u_k(t)| = \int_{k/2}^{(k+1)/2} \frac{t|q(u)|}{e^{tu}-1} du$ then $u_k(t) = (-1)^k |u_k(t)|$ for all integer $k \geq 1$, and finally establish that $$\forall n \in \mathbf{N}^*, \left|\sum_{k=n}^{+\infty} u_k(t)\right| \leq \frac{1}{2n}.$$
For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set
$$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$
Let $t \in \mathbf{R}_+$. Show successively that $|u_k(t)| = \int_{k/2}^{(k+1)/2} \frac{t|q(u)|}{e^{tu}-1} du$ then $u_k(t) = (-1)^k |u_k(t)|$ for all integer $k \geq 1$, and finally establish that
$$\forall n \in \mathbf{N}^*, \left|\sum_{k=n}^{+\infty} u_k(t)\right| \leq \frac{1}{2n}.$$