Generating Function for Fibonacci Numbers

Theorem
Let $\map G z$ be the function defined as:


 * $\map G z = \dfrac z {1 - z - z^2}$

Then $\map G z$ is a generating function for the Fibonacci numbers.

Proof
Let the form of $\map G z$ be assumed as:

where $F_n$ denotes the $n$th Fibonacci number.

Then:

and so:

Hence the result:
 * $\map G z = \dfrac z {1 - z - z^2}$