1. ## Pointwise convergence

Hi people, I am investigating the convergence of the series

$\displaystyle \lim_{k \rightarrow \infty} \sum_{j=2}^k x_j(k) = \lim_{k \rightarrow \infty} \left( \sum_{j=2}^k \frac{{k+c-j-1 \choose {k-j}}}{B(j+2a+1,a)} \right)$

where $\displaystyle c>1 \in \mathbb{N}, 0<a<1$ fixed constants and B(,) the standard beta function. Could I prove the convergence using the ratio test? eg for a large k one can use stirling's approximation to show that

$\displaystyle \lim_{k \rightarrow \infty}\frac{x_k(k)}{x_{k-1}(k)}=\frac{1}{c} < 1$

Could we continue then:
Therefore $\displaystyle (\forall ( 0< \epsilon < 1-1/c))( \mbox{ there is } N(k) \in \mathbb{N}): \forall j>N(k) \Rightarrow \frac{x_j(k)}{x_{j-1}(k)}<1-\epsilon$, etc?