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:

    “He who knows that power is inborn, that he is weak because he has looked for good out of him and elsewhere, and, so perceiving, throws himself unhesitatingly on his thought, instantly rights himself, stands in the erect position, commands his limbs, works miracles; just as a man who stands on his feet is stronger than a man who stands on his head.”
    —Ralph Waldo Emerson (1803–1882)

    “Through a series of gradual power losses, the modern parent is in danger of losing sight of her own child, as well as her own vision and style. It’s a very big price to pay emotionally. Too bad it’s often accompanied by an equally huge price financially.”
    —Sonia Taitz (20th century)