In the case of conditional convergence of a series, why do rearrangements affect the value of the series?
Rearrangements of a conditionally convergent series
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0@Henry I have never seen that definition before. I am talking about [this](http://en.wikipedia.org/wiki/Conditional_convergence). I couldn't verify your definition, but I found [this](http://en.wikipedia.org/wiki/Unconditional_convergence). I don't know if my interpretation is *the* correct intepretation, but it definitely seems like *a standard* one. – 2011-09-08
3 Answers
The intuitive answer is that in a conditionally convergent series the positive terms sum to $+\infty$ while the negative terms sum to $-\infty$. We know $\infty-\infty$ is not well defined. If we sum up the positive terms faster than the negative ones we increase the value of the sum. The link J. M. gave is a good one.
consider the series $\sum_{i=1}^\infty (-1)^{i+1}\cdot \frac1i$. It is (conditionally) convergent. We now construct a divergent rearrangement. Since $\sum_{i=1}^\infty \frac 1{2i+1}$ diverges, we can choose an increasing sequence $(N_k)$ with $N_0 = 0$ in $\mathbb N$ s.th. $\sum_{i=N_k+1}^{N_{k+1}} \frac 1{2i+1} \ge 1$ for all $k\in \mathbb N$. Now $ \sum_{k=1}^\infty \left(\sum_{i=N_k+1} \frac 1{2i+1} - \frac 1{2i}\right) $ is a divergent rearrangement of the given series. I hope, I understood your question correctly,
HTH, Yours, AB, martini.
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0*Yours* and *martini* I understand, *HTH* I think I managed to decode, but *AB*... – 2011-09-11
This is called Riemann's rearrangement theorem. A better description than I could possibly type up here is given at wikipedia. It has detailed examples and a full proof.
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2Oops, I didn't see that the link was already given by J. M. – 2011-09-08