i need to prove this
$$\sum _{n=1}^\infty \left[ \frac { p(p+1)\cdots (p+n-1) }{ q(q+1)\cdots (q+n-1) } \right]^\alpha, \qquad (p>0,q>0)$$
converges if and only if $\alpha (q-p)>1$
I tried to use Raabe test, but i dont know how to do with alpha, i supposed that it's Raabe because $\alpha (q-p)$ need to be $>1$.
Thanks for any help.
Rabee's test is a good choice here. First compute the quotient: $$\frac{a_n}{a_{n+1}} = \left[ \frac{q+n}{p+n} \right]^{\alpha}.$$ Therefore $$n \left( \frac{a_n}{a_{n+1}} - 1 \right) = \frac{ \left[ 1 + \frac{q-p}{n+p} \right]^{\alpha} - 1 }{ \frac{1}{n} } = (q-p) \cdot \frac{ \left[ 1 + \frac{q-p}{n+p} \right]^{\alpha} - 1 }{ \frac{q-p}{n+p} } \cdot \frac{n}{n+p}.$$ Now, for $f(x) = x^{\alpha}$ notice that by the definition of derivative, the middle fraction tends to $f'(1) = \alpha,$ so $$\lim_{n \to \infty} n \left( \frac{a_n}{a_{n+1}} - 1 \right) = \alpha( q-p ).$$
