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$$ \sum_{k_1=0}^{N-1}\frac{\binom{N-1}{k_1}(\beta -2)^{N-1-k_1}}{1+k_1+\alpha(N-1-k_1)} $$

where $\alpha \in \mathbb{R}$ and $\alpha >1$ (1.3 say), $\beta \in \mathbb{R}$ and $\beta >2 $, and N is a finite natural number

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    All i could do was find an upper bound for the summation as $(\beta - 1)^{N-1}$2012-01-17
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    Have you tried using Gosper's algorithm or (since it has compact support) Zeilberger's algorithm?2012-01-17
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    @PeterTaylor Had not heard of the algorithms that you have mentioned. Will read up on them and give them a try.Thanks2012-01-17
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    WolframAlpha's [answer](http://www.wolframalpha.com/input/?i=sum%28%28binomial%28n-1,k%29*%28b-2%29%5E%28n-1-k%29%29/%281%2bk%2ba*%28n-1-k%29%29,k=0..n-1%29) in hypergeometric form. ![result](http://i.stack.imgur.com/fEO3B.gif)2012-03-17

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