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Can somebody give me hint for computing following integral $$\int_1^{+\infty} \frac{dx}{x(x+1)\dots(x+k)}$$ where $k \in \mathbb{N}$. My idea was to write that as $$\frac{A_1}{x}+\dots + \frac{A_k}{x+k}$$ and integrate that but from that I haven't got anything.

smth
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  • Writing out a few $A_k$ it appears that there is a pattern – Sal Mar 27 '22 at 19:04
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    https://math.stackexchange.com/questions/114155/computing-int-0-infty-frac1x1x2-cdotsxn-mathrm-dx – Zack Mar 27 '22 at 19:08
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    https://math.stackexchange.com/questions/1066144/improper-integral-int-1-infty-frac-mathrm-dxxx1x2-cdotsxn – Zack Mar 27 '22 at 19:09

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I think if you can find a way to generalise the partial fraction decomposition that would give you the easiest form but I cannot think of a way to do it.

I do know that: $$\prod_{i=0}^k(x+i)=x(x+1)_k$$ where $(x+1)_k$ represents the rising factorial i.e. $$(x+1)_k=\frac{(x+k)!}{x!}$$ so you could represent your integral as: $$\int_1^\infty\frac{\Gamma(x)}{\Gamma(x+k+1)}\,dx$$ however I was not able to progress any further than this

Henry Lee
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