Sum of a Combinatorial Series

Question

It has been a long time since I've done combinatorics, does anybody knows how to sum this series? $$\sum_{d=0}^\infty \binom{n+d-1}d q^d$$ I'm assuming no problems of convergence since in fact is a formal series...

Answer 1

Hint. One may use the Generalized Binomial Theorem $$ \frac{1}{(1-q)^n} = \sum_{k=0}^\infty {-n \choose k}(-1)^k q^k ,\qquad |q|<1, $$ observing that $$ \binom{-n}k =\frac1{k!}\prod_{i=0}^{k-1}(-n-i) =\frac{(-1)^k}{k!}\prod_{i=0}^{k-1}(n+i) =\frac{(-1)^k}{k!}\cdot\frac{(n+k-1)!}{(n-1)!} =(-1)^k\binom{n+k-1}k $$ to get

$$ \sum_{d=0}^\infty \binom{n+d-1}d q^d=\frac{1}{(1-q)^n} ,\qquad |q|<1. $$