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In this question, a proof using real analysis is given of the following identity $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^{2}}$$

Is there a combinatorial proof of this identity? If so, does the proof require that $a$ be a natural number?

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Very good question! I'm also wondering if there are any known obstructions for an infinite-series identity to have a combinatorial proof. (Which means either a "finite partial sums" version or a "generating functions" version.) –  darij grinberg Dec 30 '12 at 1:15
You really mean "direct" instead of "combinatorial". It is a popular belief that elegant formulas must have a combinatorial proof, rather unsupported by the evidence. Negative examples include mathoverflow.net/questions/34145 and mathoverflow.net/questions/16129 –  Igor Pak Dec 30 '12 at 4:04
I believe that this identity is a limit of a finite sum that can be proved using the WZ method. Whether that makes it combinatorial is a matter of opinion. –  Ira Gessel Jul 23 '14 at 13:48

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