Generating Function for Powers of Two

Theorem
Let $\sequence {a_n}$ be the sequence defined as:
 * $\forall n \in \N: a_n = 2^n$

That is:
 * $\sequence {a_n} = 1, 2, 4, 8, \ldots$

Then the generating function for $\sequence {a_n}$ is given as:
 * $\displaystyle \map G z = \frac 1 {1 - 2 z}$ for $\size z < \dfrac 1 2$

Proof
This is valid for:
 * $\size {2 z} < 1$

from which:
 * $\size z < \dfrac 1 2$

follows directly by division by $2$.

The result follows from the definition of a generating function.