Sum of products of numbers, algebraically approaching to unity from below

Question

I would like to find the following limit for a nonnegative real number $\alpha$: $$\lim_{N\rightarrow\infty}\frac{\displaystyle\sum_{n=1}^{N-1}\prod_{m=n+1}^N\left(1-\frac{\alpha}{m}\right)}{N}.$$ Brief experiments for some $\alpha$ suggests that the answer is $\displaystyle \frac{1}{\alpha+1}$, and perhaps a series of this kind has some name in the literature, but currently I have no idea which literature I should look up or how to start my proof. Any pointers to the relevant literature or suggestions/hints are gratefully appreciated.

Answer 1

Using Pochhammer symbols$$\prod_{m=n+1}^N\left(1-\frac{\alpha}{m}\right)=\frac{\Gamma (n+1) }{\Gamma (N+1)}(n+1-\alpha)_{N-n}$$ Mathematica returns $$\displaystyle\frac 1 N\sum_{n=1}^{N-1}\prod_{m=n+1}^N\left(1-\frac{\alpha}{m}\right)= \frac{N-\alpha -\frac{\Gamma (N-\alpha +1)}{\Gamma (1-\alpha )\,\, \Gamma (N+1)}}{N(\alpha +1)}$$

Asymptotically, using Stirling approximation $$\color{blue}{\frac{1}{\alpha +1}-\frac{\alpha }{(\alpha +1) N}+ \frac{(\alpha -1)}{(\alpha +1)\,\, \Gamma (2-\alpha )\,\, N^{\alpha+1}}+\cdots}$$