   Chapter 11.6, Problem 52E

Chapter
Section
Textbook Problem

# Prove that if Σ an is a conditionally convergent series and r is any real number, then there is a rearrangement of Σ an whose sum is r. [Hints: Use the notation of Exercise 51. Take just enough positive terms a n + so that their sum is greater than r. Then add just enough negative terms a n − so that the cumulative sum is less than r. Continue in this manner and use Theorem 11.2.6.]

To determine

To prove: The sum of the rearrangement series an which is r.

Explanation

Given:

The series an is conditionally convergent and r is any real number.

Result used:

If the limnan=0 , then the series an is convergent.

Proof:

Consider series an is conditionally convergent.

That is, the series an is convergent.

Thus, by Result stated above, limnan=0 .

From the hint, the sum of the positive terms an+ and their sum an+>r the negative terms an and their sum an+<r

### 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 