A sum involving binomial coefficients and its evaluation using the Gamma function

Question

Does anyone know how to prove (or a reference for) the following identity for positive integers $r$: $$\sum_{i=0}^r (-1)^i{r\choose i}\frac{1}{ir+1}= \frac{\Gamma(1+1/r)\Gamma(r+1)}{\Gamma(r+1+1/r)}$$

Answer 1

Maple?

Input: $$f:=(-1)^i*r!/i!/(r-i)!/(r*i+1)$$ $$\text{SumTools}[\text{DefiniteSummation}](f,i=0..r)$$ Output:$${\Gamma \left( r+1 \right) \Gamma \left( {\frac {r+1}{r}} \right) \left( \Gamma \left( {\frac {{r}^{2}+r+1}{r}} \right) \right) ^{-1}} $$ Maple uses the algorithms of Gosper, Wilf, Zeilberger and others using telescoping sums. It should be able to convert the computation into a human readable proof, a so-called certificate.