   Chapter 11.6, Problem 2E

Chapter
Section
Textbook Problem

Determine whether the series is absolutely convergent or conditionally convergent. ∑ n = 1 ∞ ( − 1 ) n − 1 n

To determine

Whether the given series is absolutely convergent or conditionally convergent.

Explanation

1) Concept:

Use the alternating series test, definition of absolutely convergent series and conditionally convergent series.

2) Alternating series test:

If the alternating series

n=1-1n-1bn=b1-b2+b3-b4+b5-b6+         bn>0

satisfies

(i)  bn+1bn for all n

(ii) limnbn=0

then the series is convergent.

3) Definition:

a) Absolutely convergent series:

A series Σ an is called absolutely convergent when the series of absolute values Σ|an| is convergent.

b) Conditionally convergent series:

A series Σ an is called conditionally convergent when it is convergent but not  absolutely convergent.

4) Given:

n=1-1n-1n

5) Calculation:

Consider.

n=1-1n-1bn=n=1-1n-1n

Hence,bn=1n

Also bn=1n>0  for  n1

For all n1,    n+1>n , i

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 