What is the partial sum?

Question

Let $0<\alpha<1, a_n= \frac{n!}{\alpha(\alpha+1)(\alpha+2)..(\alpha+n-1)}$.

Is it possible to write the partial sum $\sum_{k=0}^{n}a_k$ in a compact form?

Thanks in advance.

Answer 1

Put $b_k=\frac{(k+1)!}{\alpha(\alpha+1)\cdots(\alpha+k-1)}$, and compute $b_k-b_{k-1}$.