TITLE: The convergence of a sequence with infinite products
QUESTION [5 upvotes]: I have a problem to determine convergence (sum over n). $$\sum_{n=0}^\infty \dfrac {a\left( a+1^{p}\right) \ldots \left( a+n^{p}\right) }{b\left( b+1^{p}\right) \ldots \left( b+n^{p}\right) }$$where $a<b, a>0,b>0$.
I have concluded convergence for $p\leq0$ by comparing it to a constructed geometric sequence, as well as for $p=1$, using comparison test with $n^{a-b}$. But I can not use similar methods for $p>1$ and $0<p<1$. 
I have some thoughts for the two parts:
When $p>1$, it seems that the limit of each term is not $0$. If the limit could be evaluated, then the divergence can be proved. My method for $p=1$ is to use the Euler Product of the gamma function, but the $p$ power makes it impossible to use this method. I am wondering if there is any kind of generalization of gamma function that is of this form.
when $0<p<1$, I compared it to the case of $p=1$, that could at least tell it converges in the range when $p=1$ converge. But it is inconclusive for the parts remaining.  
Any help or hints would be appreciated.

REPLY [2 votes]: To me this is a simple application of Raabe-Duhamel's test. You'll get convergence for $p<1$ and, if $b-a>1$, you also get convergence for $p=1$. For $p>1$ the series diverges.