Closed form expression for $\sum_{n=0}^\infty H(x - n y) z^n$ involving Heaviside functions

Question

For $x>0$, $y>0$ and $0 < z < 1$ consider the sum $$ \sum_{n=0}^\infty H(x - n y) z^n\ , $$ where $H$ is the Heaviside step function. Is there a way to write down a closed form expression for this?

Answer 1

This is the sum of a finite geometric progression: $$\sum_{n:x\geq yn,n\geq0}{z^n}=\sum_{n=0}^{\left\lfloor\frac{x}{y}\right\rfloor}{z^n}=\frac{z^{\left\lfloor\frac{x}{y}\right\rfloor+1}-1}{z-1}$$