   Chapter 11.9, Problem 12E

Chapter
Section
Textbook Problem

# Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence.12. f ( x ) = 2 x + 3 x 2 + 3 x + 2

To determine

To express: The function as the sum of a power series by using partial fractions and find the interval of convergence.

Explanation

Given:

Let f(x)=2x+3x2+3x+2

Factorize the function as follows:

2x+3x2+3x+2=2x+3(x+1)(x+2)

Apply partial fraction method.

2x+3(x+1)(x+2)=A(x+1)+B(x+2)=Ax+1×x+2x+2+Bx+2×x+1x+1=A(x+2)(x+1)(x+2)+B(x+1)(x+1)(x+2)=A(x+2)+B(x+1)(x+1)(x+2)

Therefore the denominators of both sides are equal and also the numerator must be equal.

2x+3=A(x+2)+B(x+1) (1)

Find the value of A and B.

Substitute x=1 in equation (1)

2(1)+3=A((1)+2)+B((1)+1)2+3=A(1)+B(0)A=A+0A=1

Substitute x=2 and A=1 in equation (1),

2(2)+3=A((2)+2)+B((2)+1)4+3=A(0)+B(1)1=0BB=1

Substitute the values in the function.

f(x)=Ax+1+Bx+2=1x+1+1x+2=p(x)+q(x)

Find the power series for p(x)=11+x

The function can be written as follows:

p(x)=11+x=11(x)=a1r

Therefore, p(x) is a sum of a geometric series with initial term a=1 and common ratio r=x .

p(x)=n=0arn=n=0(1)(x)n=n=0(1)n(x)n

Find the power series for q(x)=1x+2

Divide the numerator and denominator by 2.

q(x)=12x21

The function can be written as follows:

12x2+1=121(x2)=a1r

Therefore, q(x) is a sum of a geometric series with initial term a=12 and common ratio r=x2

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