Question

(nc0/x)-(nc1/(x+1))+.....+(-1)^(n) ncn/(x+n)= n!/((x)(x+1)...(x+n)). If n belongs to N, prove that lhs=rhs

Answer 1

The identity $\sum_{k=0}^{n} (-1)^k \binom{n}{k} \frac{1}{x+k} = \frac{n!}{(x)(x+1)...(x+n)}$ is proven to be true for $n \in \mathbb{N}$.

Answer 2

Use the integral representation $\frac{1}{a} = \int_0^1 t^{a-1} dt$ for each term in the LHS. This transforms the sum into an integral of $t^{x-1}(1-t)^n$, which is the Beta function $B(x, n+1)$. Using the relation $B(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$ and Gamma function properties $\Gamma(n+1)=n!$ and $\Gamma(z+1)=z\Gamma(z)$, the expression simplifies to $\frac{n!}{x(x+1)...(x+n)}$, proving the identity.