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 :
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:
“With an eye made quiet by the power
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We see into the life of things.”
—William Wordsworth (17701850)
“Life ... is not simply a series of exciting new ventures. The future is not always a whole new ball game. There tends to be unfinished business. One trails all sorts of things around with one, things that simply wont be got rid of.”
—Anita Brookner (b. 1928)