Generating Function for Powers of Two

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

That is:
 * $\left \langle {a_n}\right \rangle = 1, 2, 4, 8, \ldots$

Then the generating function for $\left \langle {a_n}\right \rangle$ is given as:
 * $\displaystyle G \left({z}\right) = \frac 1 {1 - 2z}$ for $\left|{z}\right| < \dfrac 1 2$

Proof
Follows directly from Sum of Infinite Geometric Progression:

This is valid for:
 * $\left|{2 z}\right| < 1$

from which:
 * $\left|{z}\right| < \dfrac 1 2$

follows directly by division by $2$.

The result follows from the definition of a generating function.