Fibonacci Numbers - Power Series

Power Series

The generating function of the Fibonacci sequence is the power series

This series has a simple and interesting closed-form solution for :

This solution can be proven by using the Fibonacci recurrence to expand each coefficient in the infinite sum defining :

\begin{align} s(x) &= \sum_{k=0}^{\infty} F_k x^k \\ &= F_0 + F_1x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k \\ &= x + \sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \\ &= x + x\sum_{k=0}^{\infty} F_k x^k + x^2\sum_{k=0}^{\infty} F_k x^k \\ &= x + x s(x) + x^2 s(x). \end{align}

Solving the equation for results in the closed form solution.

In particular, math puzzle-books note the curious value, or more generally

for all integers .

More generally,

Read more about this topic:  Fibonacci Numbers

Famous quotes containing the words power and/or series:

    “Today, supremely, it behooves us to remember that a nation shall be saved by the power that sleeps in its own bosom; or by none; shall be renewed in hope, in confidence, in strength by waters welling up from its own sweet, perennial springs. Not from above; not by patronage of its aristocrats. The flower does not bear the root, but the root the flower.”
    —Woodrow Wilson (1856–1924)

    “In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
    —Jorge Luis Borges (1899–1986)