Suppose $0 < \theta < 1$ and $c$ is a positive constant no more than 3. I want to find a sharp lower bound for the series $$\sum_{k = 1}^{\infty} \frac{k (1 - \theta)^k}{k! \log^3 (c + k)}.$$ Since the denominator contains the log function, I don't know how to deal with it.
Does anyone know the solution? Thanks so much!