   Chapter 11.10, Problem 11E

Chapter
Section
Textbook Problem

# Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence.11. f(x) − (1 − x)−2

To determine

To find: The Maclaurin series for f(x) by using the definition of a Maclaurin series and also the radius of the convergence.

Explanation

Given:

The function is, f(x)=(1x)2 .

Result used:

(1) The expansion of the Maclaurin series f(x)=n=0f(n)(0)n! is,

f(0)+f01!x+f(0)2!x2+f(0)3!x3+

(2) The Ratio Test:

(i) If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent (and therefore convergent.)

(ii) If limn|an+1an|=L>1 or limn|an+1an|= , then the series n=1an is divergent.

(ii) If limn|an+1an|=1 , the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an .

Calculation:

Obtain f(0) .

Substitute 0 for x in f(x) .

f(0)=(10)2=(1)2=1(1)2=1

Find the first derivative of f(x) at a=0 .

f(x)=ddx(1x)2=(2)(1x)21ddx(1x)=2(1x)3(1)

f(x)=2(1x)3 (1)

Obtain f(0) .

f(0)=2(10)3=2(1)3=2

Find the second derivative of f(x) at a=0 .

f(2)(x)=d2dx2(f(x))=ddx(f(x))=ddx(2(1x)3)    (by equation (1))=2(3)(1x)4ddx(1x)

Simplify further and obtain f(2)(0) as shown below.

f(2)(x)=6(1x)4 (2)

Substitute 0 for x,

f(2)(0)=6(10)4=6(1)4=6

Find the third derivative of f(x) at a=0 .

f(3)(x)=d3dx3(f(x))=ddx(f(2)(x))=ddx(6(1x)4)    (by equation(2))=6(4)(1x)5ddx(1x)

Simplify further and obtain f(3)(0) .

f(3)(x)=24(1x)5(1)

f(3)(x)=24(1x)5 (3)

Substitute 0 for x,

f(3)(0)=24(10)5=24(1)=24

Find the fourth derivative of f(x) at a=0

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