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I think understand that when I have a conditionally convergent series, it consists of series of positive and negative values which are divergent and thus one can find such permutation of indices $\phi : \mathbb{N}\rightarrow\mathbb{N}$ so the rearranged series sums up to an arbitrary value, diverges or oscillates.

This is how I understand what Riemann's rearrangement theorem says, but how do I use it practically? For example, when I have the conditionally convergent series:

$$\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}$$

and I want to rearrange it to be divergent or to sum up to certain value M? How do I define such bijections $\phi$ ?

I appreciate all help.

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The proof of Riemann's rearrangement theorem is "constructive" in the sense that it tells you how to go about constructing the rearranged series; from there you can describe the bijection $\phi$. –  Arturo Magidin Jun 19 '11 at 2:19

2 Answers 2

up vote 8 down vote accepted

I don't know if I can give you a formula, but I can give you an algorithm. First take enough positive terms to make the sum exceed $M$; then enough negative terms to get below $M$; then more positive terms to get above $M$, more negative terms to get below, etc.

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It is described here in detail how any real number can be obtained as sum of a rearranged series of the alternating harmonic series.

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