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?

Assuming the series converges, we have 
\begin{align}
& \sum_{i = 0}^\infty \rho^i \prod_{j = 1}^i \left(\frac{\alpha}{j} + 1\right)\\
&= \sum_{i = 0}^\infty \rho^i \prod_{j = 1}^i \frac{\alpha + j}{j}\\
&= \sum_{i = 0}^\infty \rho^i \frac{(\alpha + 1)(\alpha + 2)\cdots (\alpha + i)}{i!}\\
&= \sum_{i = 0}^\infty (-1)^i \rho^i \frac{(-\alpha - 1)(-\alpha - 2)\cdots (-\alpha - i)}{i!}\\
&= \sum_{i = 0}^\infty (-\rho)^i \binom{-\alpha-1}{i}\\
&= (1-\rho)^{-\alpha-1}\\
&= \frac{(1-\rho)^{-\alpha}}{1-\rho}.
\end{align}
