Generating Function for Sequence of Powers of Constant

Theorem
Let $c \in \R$ be a constant.

Let $\left \langle {a_n}\right \rangle$ be the sequence defined as:
 * $\forall n \in \Z_{\ge 0}: a_n = c^n$

That is:
 * $\left \langle {a_n}\right \rangle = 1, c, c^2, c^3, \ldots$

Then the generating function for $\left \langle {a_n}\right \rangle$ is given as:
 * $G \left({z}\right) = \dfrac 1 {1 - c z}$

Proof
Consider the sequence $\left \langle {b_n}\right \rangle$ defined as:
 * $\forall n \in \Z_{\ge 0}: a_n = 1$

Let $H \left({z}\right)$ be the generating function for $\left \langle {b_n}\right \rangle$.

Then:

The result follows from the definition of a generating function.