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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? Is so, does the proof require that $a$ be a natural number? Also is there an easy way to verify if combinatorial proofs exist of particular identities?

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    $\begingroup$ I can answer the final question: No. $\endgroup$ – Chris Godsil Dec 29 '12 at 5:05

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