A sort of modified geometric series

Question

I was wondering if there are any hints on how to manage this series $$ \sum_{i=2}^\infty \prod_{j=1}^{i-1} (1-cj^{\beta-1})=(1-c)+(1-c)(1-c2^{\beta-1})+(1-c)(1-c2^{\beta-1})(1-c3^{\beta-1})... $$ with $0<c<1$ and $0\leq \beta\leq 1$, which for $\beta=1$ reduces to the geometric series.