   Chapter 11.9, Problem 5E

Chapter
Section
Textbook Problem

# Find a power series representation for the function and determine the interval of convergence.5. f ( x ) = 2 3 − x

To determine

To find: The power series representation for the function f(x)=23x and determine the interval of convergence.

Explanation

Given:

The function is f(x)=23x .

Result used:

(1) “The sum of the geometric series with initial term a and common ratio r is n=0arn=a1r . (1)

(2) 11x=1+x+x2+x3+=n=0xn (2)

Calculation:

Consider the function f(x)=23x

Divide the numerator and denominator by 3.

21x=231x3=2311(x3)

Substitute x3 for x in equation (2),

11x=11(x3)=1+(x3)+(x3)2+(x3)3+...=n=0(x3)n

Therefore, 21x=23n=0(x3)n

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