Closed form for infinite series $\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)\ldots(n+m)}$

Asked Jan 01 '19 at 18:25

Active Jan 01 '19 at 18:37

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Let $m$ be a positive integer.
Is there a closed form for the infinite series $\displaystyle\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)\ldots(n+m)}$?

In the cases $m=1, 2$, one can use a partial fraction decomposition to make the sum telescope. However this seems to become much more difficult for larger $m$ (and I'm not sure whether it continues to telescope). I suspect there is a closed form, but I don't know how one would obtain it.

edited Jan 01 '19 at 18:37

amWhy

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asked Jan 01 '19 at 18:25

AlephNull

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It still telescopes. – Angina Seng Jan 01 '19 at 18:26
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I couldn't find this question when I searched before posting it; apologies for the duplicate. – AlephNull Jan 01 '19 at 18:36
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Here is another reference that might be helpful. https://math.stackexchange.com/questions/1824603/evaluate-frac04-frac15-frac26-frac37-frac48 – Larry Jan 01 '19 at 18:37

Closed form for infinite series $\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)\ldots(n+m)}$

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