Generating Function for Sequence of Powers of Constant

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

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

That is:
 * $\sequence {a_n} = 1, c, c^2, c^3, \ldots$

Then the generating function for $\sequence {a_n}$ is given as:
 * $\map G z = \dfrac 1 {1 - c z}$

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

Let $\map H z$ be the generating function for $\sequence {b_n}$.

Then:

The result follows from the definition of a generating function.