Fibonacci Number - 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,

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Famous quotes containing the words power and/or series:

“Do not say to yourself, “My power and the might of my own hand have gotten me this wealth.” But remember the LORD your God, for it is he who gives you power to get wealth, so that he may confirm his covenant that he swore to your ancestors, as he is doing today.”
—Bible: Hebrew, Deuteronomy 8:17,18.

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—Jean Baudrillard (b. 1929)