2x3 18. (a) Use the geometric series to derive a power series for the function f(x) = 2+x (b) What is the radius of convergence?

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question 18

2x3
18.
(a) Use the geometric series to derive a power series for the function f(x) =
2+x
(b) What is the radius of convergence?
Transcribed Image Text:2x3 18. (a) Use the geometric series to derive a power series for the function f(x) = 2+x (b) What is the radius of convergence?
Expert Solution
Step 1

Given,

The function 

Calculus homework question answer, step 1, image 1

 

Step 2

The formula of geometric power series is

Calculus homework question answer, step 2, image 1

Where a is first term and r is the common ratio.

Remember that,

For |r| ≥ 1 , the series diverges.

|r|< 1 , the series converges.

 

By using this result, plug  a =1  and r = x gives,

Calculus homework question answer, step 2, image 2

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