   Chapter 11.8, Problem 16E

Chapter
Section
Textbook Problem

# Find the radius of convergence and interval of convergence of the series.16. ∑ n = 1 ∞ ( − 1 ) n ( 2 n − 1 ) 2 n ( x − 1 ) n

To determine

To find: The radius of convergence and interval of convergence of the series.

Explanation

Given:

The series n=1(1)n(2n1)2n(x1)n .

Result used:

(1)Ratio test:

If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent.

(2) The p-series n=11n is converges if p>1 and diverges if p1 .

(3) Limit comparison test:

“Suppose that an and bn are the series with positive terms, if limnanbn=c , where c is a finite number and c>0 , then either both series converge or both diverge.”

(4) Alternating Series Test:

“If the alternating series n=1(1)n1bn=b1b2+b3b4+...   bn>0 satisfies, (i) bn+1bn   for all n and (ii) limnbn=0 , then the series is convergent”

Calculation:

Let an=(1)n(2n1)2n(x1)n .

Then, an+1=(1)n+1(2(n+1)1)2n+1(x1)n+1 .

Obtain |an+1an| to apply the Ratio test.

|an+1an|=|(1)n+1(2(n+1)1)2n+1(x1)n+1(1)n(2n1)2n(x1)n|

Take limn on both sides,

limn|an+1an|=limn|(1)n+1(2(n+1)1)2n+1(x1)n+1(1)n(2n1)2n(x1)n|=limn|(1)n+1(2n+1)2n+1(x1)n+1.(2n1)2n(1)n(x1)n|=limn|(1)n+1(1)n.(2n1)(2n+1).(x1)n+1(x1)n.2n2n+1|=limn[2n12n+1|x1|2]

Simplify the terms further,

limn|an+1an|=limn[21n2+1n.|x1|2]

Apply the limit and simplify the terms as shown below.

limn[21n2+1n.|x1|2]=[212+1.|x1|2]=|x1|2(202+0)=|x1|2

The series n=1(1)n(2n1)2n(x1)n converges as |x|<1

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Differentiate. y=sint1+tant

Single Variable Calculus: Early Transcendentals, Volume I

#### Expand each expression in Exercises 122. (x32x2+4)(3x2x+2)

Finite Mathematics and Applied Calculus (MindTap Course List)

#### In Exercises 31-34, evaluate h(2), where h = g f. 34. f(x)=1x1;g(x)=x2+1

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### The graph of x = cos t, y = sin2 t is:

Study Guide for Stewart's Multivariable Calculus, 8th 