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Conditionally Convergent Series

Prove that if Series An (small "a", sub "n") is a conditionally convergent series and r is any real number, then there is a rearrangement of Series An whose sum is r. [Hints: Use the notation of Exercise 39 (I'll show below). Take just enough positive terms An+ so that their sum is greater than r. Then add just enough negative terms An- so that the cumulative sum is less than r. Continue in this manner and use Thereom 11.2.6 (Shown below).

Thereom 11.2.6 - If the Series An ( n=1 to infiniti) is convergent, then Lim (n goes to infiniti) An = 0.

Number 39 to be used to solve for number 40 - Given any series An, we define a series An+ whose terms are all the positive terms of Series An and a Series An- whose terms are all the negative terms of Series An. To be specific, we let:

An+ = {An + An (absolute value)}/2

An- = {An - An (absolute value)}/2

Notice that if An > 0, then An+ = An and An- = 0, whereas if An < 0, then An- =An and An+ = 0.

Note to problem solver: All of the "A"s above really are lowercase "a"s.
What is trying to be solved is the proof in the first paragraph. All of the other information is there to help solve the problem.

Additionally, whenever you see the word "Series" it's really that funky looking E thing. I didn't know how to type that out.

Solution Summary

A Conditionally Convergent Series is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.