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,

Read more about this topic:  Fibonacci Number

Famous quotes containing the words power and/or series:

    “Mothers who have little sense of their own minds and voices are unable to imagine such capacities in their children. Not being fully aware of the power of words for communicating meaning, they expect their children to know what is on their minds without the benefit of words. These parents do not tell their children what they mean by “good” much less why. Nor do they ask the children to explain themselves.”
    —Mary Field Belenky (20th century)

    “Galileo, with an operaglass, discovered a more splendid series of celestial phenomena than anyone since.”
    —Ralph Waldo Emerson (1803–1882)