(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the series [tex]\displaystyle\Sigma_{n=1}^{\infty} \frac{(-1)^n}{n}[/tex] is not absolutely convergent. Do so by permuting the terms of the series one can obtain different limits.

2. Relevant equations

3. The attempt at a solution

I don't have a total solution; because I am not familiar with the terminology of "permuting".

I assume (for example): a permute [tex]\pi_1 = \{1,3,5,7...\}[/tex] all of the odd values. And another permute [tex]\pi_2 = \{2,4,6,8...\}[/tex] the even values. You could show that...

[tex]\displaystyle\Sigma_{\pi_1(n)}^\infty \frac{(-1)^n}{n} \to x[/tex]

Where as

[tex]\displaystyle\Sigma_{\pi_2(n)}^\infty \frac{(-1)^n}{n} \to y \neq x[/tex]

Is that on the right track?

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# Homework Help: Permuting the Terms

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