Test the series for convergence if $\ \beta-\alpha \neq 1$

Question

Test the series for convergence, if $\ \beta-\alpha \neq 1$,

$$1+\frac{1+\alpha}{1+\beta}+\frac{(1+\alpha)(2+\alpha)}{(1+\beta)(2+\beta)}+\frac{ (1+\alpha)(2+\alpha)(3+\alpha)}{(1+\beta)(2+\beta)(3+\beta)}+\ldots$$

I tried Ratio test, but it gives me $1$, which doesn't prove anything. Any hint?

Answer 1

if $\ \beta-\alpha \neq 1$, let the difference be equal to zero which is not equal to 1, and then you will get the series 1 +1 +1 +... which is a divergent series