summation of a infinite series i want to know if this infinite sum $\sum_{n=1}^{\infty}$ $\frac{a(a+1)(a+2)......(a+n-1)}{b(b+1)(b+2)......(b+n-1)}$ converges or diverges ? where a>0 and b>a+1.
if the sum converges what is the sum i.e where it will converge? i need a concrete explanation
i found some inequalities $\frac{a(a+1)}{b(b+1)}$ <$\frac{a}{b+1}$ and $\frac{a(a+1)(a+2)}{b(b+1)(b+2)}$< $\frac{a}{b+2}$ .......and continuing $\frac{a(a+1)....(a+n-1)}{b(b+1)....(b+n-1)}$ < $\frac{a}{b+n-1}$
this inequalities can be useful for this problem

Let $u_n=\dfrac{a(a+1)\ldots(a+n-1)}{b(b+1)\ldots(b+n-1)}$
We have $\dfrac{u_{n+1}}{u_n}=1-\dfrac{b-a}{n}+o(\dfrac{1}{n})$
Hence by Raabe Duhamel  :
If $b-a>1$ the infinite sum converge
If $b-a<1$ the infinite sum diverge.
