Pointwise convergence

**ory** · November 9th 2008, 02:54 PM

Hi people, I am investigating the convergence of the series

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

where $c>1 \in \mathbb{N}, 0<a<1$ 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

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

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

Thanks in advance

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