   Chapter 11.P, Problem 24P

Chapter
Section
Textbook Problem

(a) Show that the Maclaurin series of the function f ( x ) = x 1 − x − x 2 is ∑ n = 1 ∞ f n x n where f n , is the nth Fibonacci number, that is, f 1 = 1 , f 2 = 1 , and f n = f n − 1 + f n − 2 for n ≥ 3 .[Hint: Write x / ( 1 − x − x 2 ) = c 0 + c 1 x + c 2 x 2 + ... and multiply both sides of this equation by 1 − x − x 2 .](b) By writing f ( x ) as a sum of partial fractions and thereby obtaining the Maclaurin series in a different way, find an explicit formula for the nth Fibonacci number.

To determine

(a)

To show:

The Maclaurin series of fx  is

n=1fnxn

Explanation

1) Concept:

Fibonacci number:

fn=fn-1+fn-2, n3, where, f1=1 &f2=1

2) Given:

fx=x1-x-x2

3) Calculation:

Consider,

x1-x-x2=c0+c1x+c2x2+

x=1-x-x2c0+c1x+c2x2+

x=c0+c1x+c2x2+-xc0+c1x+c2x2+-x2c0+c1x+c2x2+

x=c0+c1x+c2x2+-xc0-c1x2-c2x3--x2c0-c1x3-c2x4-

x=c0

To determine

(b)

To show:

The explicit formula for the n th Fibonacci number

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