How to evaluate the infinite series $\sum _{i=0}^{\infty } \rho ^i \prod _{j=1}^i \left(\frac{\alpha }{j}+1\right)$

Mathematica shows that the infinite series

\begin{align}\sum _{i=0}^{\infty } \rho ^i \prod _{j=1}^i \left(\frac{\alpha }{j}+1\right)= -\frac{(1-\rho )^{-\alpha }}{\rho -1}. \end{align}

How do I prove this?

• Starting from the rhs, using the binomial expansion and long division would show it. But, I suppose that you want to start from the lhs, isn't it ? – Claude Leibovici Jan 16 '15 at 11:04