   Chapter 11, 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 (b) where fn is the nth Fibonacci number, that is, f1 = 1, f2 = 1, and fn = fn−1 + fn−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 − x2.] (c) 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.

(a)

To determine

To show: The Maclaurin series of the function.

Explanation

Given:

The function is f(x)=x1xx2

Calculation:

Consider the, f(x)=x1xx2 it can be written as,

x1xx2=n=0cnxnx1xx2=c0+c1x+c2x2+c3x3+x=(1xx2)(c0+c1x+c2x2+c3x3+)x=(c0+c1x+c2x2+c3x3+c4x4+c5x5+c0xc1x2c2x3c3x4c4x5c0x2c1x3c2x4c3x5)

Further simplification,

x=(c0+c1x+c2x2+c3x3+c4x4+c5x5+c0xc1x2c2x3c3x4c4x5c0x2c1x</

(b)

To determine

To find: The explicit formula for the nth Fibonacci number.

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