Euler-Binet Formula/Proof 4

Proof
From Generating Function for Fibonacci Numbers, a generating function for the Fibonacci numbers is:


 * $G \left({z}\right) = \dfrac z {1 - z - z^2}$

Hence:

where:
 * $\phi = \dfrac {1 + \sqrt 5} 2$
 * $\hat \phi = \dfrac {1 - \sqrt 5} 2$

By Sum of Infinite Geometric Progression:


 * $\dfrac 1 {1 - \phi z} = 1 + \phi z + \phi^2 z^2 + \cdots$

and so:


 * $G \left({z}\right) = \dfrac 1 {\sqrt 5} \left({1 + \phi z + \phi^2 z^2 + \cdots - 1 - \hat \phi z - \hat \phi^2 z^2 - \cdots}\right)$

By definition, the coefficient of $z^n$ in $G \left({z}\right)$ is exactly the $n$th Fibonacci number.

That is:
 * $F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$