Calculus: Analytic Geometry and Calculus, with Vectors

626

Hint: Start by writing

1+2x

1-x-x2 = ao + a1x + a2 + a3X3 +

and

1 + 2x = ao + alx + a2x2 + a3x3 + aax4 +

-aox-aix2-aax3-a3x4-

-aox2-a1x3-aax4-

Then obtain the formulas in the first column

1 = ao

2=a1-ao

0 = a2- al -a0

0 = a3-a2-a1

Series

ao1

a13

a2 = 4

as = 7

which determine the answers in the second column.

16 The sequences

(1) 1, 1, 2, 3, 5, 8, 13, 21,

(2) 1, 3, 4, 7, 11, 18, 29, 47,

are examples of Fibonacci sequences, that is, sequences for which each element after

the first two is the sum of its two nearest predecessors. A little work with power

series reveals some surprising information about these famous sequences. Let

Fo, F1, F2, be a Fibonacci sequence. Letting g be defined by the first of

the formulas

(3) g(x) = Fo + Fix + Fax2 + Fax3 + 1.4x4 +

(4) xg(x) = Fox + F1x2 + Faxa + Fax4 +

(5) xsg(z) = Fox2 + Fix' + F2x4 +

tell how the next two are obtained. Subtract the last two formulas from the

first and use the result to show that

(6) g(x) =

Fo + (F1 - Fo)x

1 -x-x2

New and illuminating formulas are obtained by expressing g(x) as asum of par-

tial fractions and expanding these fractions into power series. To simplify

writing, let

(7) J =2- 1= 0.618034, B =2+ 1= 1.618034.

Observe that r1B = 1. Show (the details are a bit onerous) that

(8)

(^1) -x-x2-[1-Bx+1B A
} fix]
and hence that, when IBxj < 1,
(9) (^1) 2 = 1 [Bnt1 + (-1)nAnt1]xn.
1-x-x n=a5