I need estimate the following sum:
$\sum_{d=1}^{n}\frac{\mu(d)}{d}\sum_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}$, where $q>1$ and $\mu$ is the Möbius function.
To obtain the main term, I need to find the following sum:
$q+\frac{1}{2}q^2+\cdots+\frac{1}{n}q^n$. Though the second problem looks like calculus problem, I still have no good control about it.