[Math] Prove properties of the factorial (gamma function)

Question

I want to prove the equation is satisfied. 'p' is a natural number.
$$\sum_{n=1}^{∞} \frac{\Gamma(n)}{\Gamma(n+p+1)}=\frac{1}{p^2\Gamma(p)}$$
Understandably, This formula can be written as this.
$$\sum_{n=1}^{∞} \frac{1}{n(n+1)\cdots(n+p)}=\frac{1}{pp!}$$
Please give me hint or prove.

Answer 1

The Beta function to the rescue:

$$\frac{\Gamma(n)\Gamma(p+1)}{\Gamma(n+p+1)} = B(n,p+1) = \int_0^1 t^{n-1}(1-t)^p\,dt.$$