I am trying to caclulate the Factorial Moment of the Geometric Distribution #2 with parameter $p$. Therefore I set $\Omega = \mathbb{N}_0$ and have by using the pochhammer symbol and setting $q=1-q$ that
$E((k)_l)= \sum _{k=0}^{\infty } (k)_l p q^k = p^{-l} q \cdot l! \sum _{k=0}^{\infty } (\frac{(k+l-1)!}{(k-1)! \cdot l!}\cdot p^{l+1} q^{k-1}) $
Now Mathematica tells me that $\sum _{k=0}^{\infty } (\frac{(k+l-1)!}{(k-1)! \cdot l!}\cdot p^{l+1} q^{k-1})=1$, but I cannot see why this identity is true. Also when using
FactorialMoment[GeometricDistribution[p], l]
Mathematica suggests that $E((k)_l)=(\frac{q}{p})^l l!$. Thank you in Advance for your help.