How to prove the below equation?

Revision en3, by yspm, 2021-10-13 05:22:47

I found the below equation when studying probabilities:

$$$n\binom{n-1}{k-1}\sum\limits_{i=0}^{n-k}\binom{n-k}i (-1)^i\frac{1}{k+i+1}=\frac{k}{n+1}$$$

(Here $$$n,k$$$ are positive integers and $$$n\ge k$$$)

but failed to prove it.

Can anyone offer precious help to me. Thanks

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  Rev. Lang. By When Δ Comment
en3 English yspm 2021-10-13 05:22:47 21 Tiny change: 'ies:\n\n$$\sum\limit' -> 'ies:\n\n$$n\binom{n-1}{k-1}\sum\limit'
en2 English yspm 2021-10-13 03:35:07 51
en1 English yspm 2021-10-13 03:21:25 243 Initial revision (published)