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:

    “The poet is the person in whom these powers are in balance, the man without impediment, who sees and handles that which others dream of, traverses the whole scale of experience, and is representative of man, in virtue of being the largest power to receive and to impart.”
    —Ralph Waldo Emerson (1803–1882)

    “I thought I never wanted to be a father. A child seemed to be a series of limitations and responsibilities that offered no reward. But when I experienced the perfection of fatherhood, the rest of the world remade itself before my eyes.”
    —Kent Nerburn (20th century)