Something nice that came across with:
Is the product $\displaystyle \prod_{n=0}^{\infty}\frac{\beta +n}{\alpha+n }$ converge or diverge?
Nice. Obvious when $\displaystyle \alpha=\beta$, when these are not equal taking logs and considering the convergence of the resulting series seems to work a treat.
(If $\displaystyle \alpha$ and/or $\displaystyle \beta $ are/is negative a bit more care is needed but it still seems to work)
CB