This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to rewrite $\alpha $ as $a+1$ and $\beta$ as $b+1$ to make $S= \frac{b!}{a!} \sum\limits_{n \ge 1} \frac{(a+n)!}{(b+n)!}$ which looks better. It is easy to prove the convergence via the ratio test but finding the exact sum is challenging.
This section of problems in the book is all about telescopic sums which I couldn't figure out how to do the decomposition of each term