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:

$\map G z = \dfrac 1 {1 - 2 z}$ for $\size z < \dfrac 1 2$

$\ds $

$\ds 1 + 2 z + 4 z^2 + \cdots$

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} \paren {2 z}^n$

$\ds $

$=$

$\ds \frac 1 {1 - 2 z}$

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.

$\blacksquare$