It can be proved that the terms of any conditionally convergentseries can be rearranged

Chapter 9, Problem 51

(choose chapter or problem)

It can be proved that the terms of any conditionally convergent series can be rearranged to give either a divergent series or a conditionally convergent series whose sum is any given number \(S\) For example, we stated in Example 2 that

\(\ln 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots\)

Show that we can rearrange this series so that its sum is \(\frac{1}{2} \ln 2\) by rewriting it as

\(\left(1-\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{3}-\frac{1}{6}-\frac{1}{8}\right)+\left(\frac{1}{5}-\frac{1}{10}-\frac{1}{12}\right)+\cdots\)

[Hint: Add the first two terms in each grouping.]

Equation Transcription:

Text Transcription:

S

ln 2 = 1 - frac{1}{2} + frac{1}{3} - frac{1}{4} + frac{1}{5} - frac{1}{6} + cdot cdot cdot

frac{1}{2} ln 2

(1- frac{1}{2} - frac{1}{4}) + (frac{1}{3} - frac{1}{6} - frac{1}{8}) + (frac{1}{5} - frac{1}{10} - frac{1}{12}) + cdot cdot cdot