   Chapter 11.8, Problem 5E

Chapter
Section
Textbook Problem

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

To determine

To find:

The radius of convergence and the interval of convergence of the series n=1xn2n-1

Explanation

1) Concept:

i) For a power series n=0cnx-an, there is a positive number R such that the series converges if x-a<R and diverges if x-a>R, this number R is called as a radius of convergence. From this, there are four possible cases of interval of convergence

a-R, a+R,  a-R, a+R,   a-R, a+R,  a-R, a+R

ii) Alternating series:

n=1-1n-1bn Converges, if bn+1bn and limnbn=0

iii) The Comparison Test:an and bn are series with positive terms.

If bn is convergent and anbn for all n, then an is also convergent.

If bn is divergent and anbn for all n, then an is also divergent.

iv) The ratio test states that if limnan+1an<1, then the series n=1an converges.

2) Given:

n=1xn2n-1

3) Calculation:

The given series is n=1xn2n-1

Therefore, the nth term is an=xn2n-1

Therefore,

limnan+1an

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

In Exercises 1-6, simplify the expression. 3+(1)2

Calculus: An Applied Approach (MindTap Course List)

True or False: is a geometric series.

Study Guide for Stewart's Multivariable Calculus, 8th

Estimate 14(x3x2) using the Left Endpoint Rule and n = 5. a) 203 b) 20 c) 24 d) 30

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Explain how to complete the square on x217x.

College Algebra (MindTap Course List) 