Upper bound for an infinite series of Pochhammer Symbol

Question

Let $a_n = \frac{1}{n!}\prod_{i=0}^{n-1} (r+\alpha i)$, for constants $0<r, \alpha<1$. The series is convergent by the ratio test. I want to find the exact value or maybe an upper bound for the infinite series $$ \sum_{n=0}^{\infty} n\,a_n $$ in terms of $r$ and $\alpha$. I appreciate any help!

Answer 1

The sum is $r/(1-\alpha)^{(1+r/\alpha)}$ by the binomial theorem.