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Learning Series for the first time and was wondering if any two series which both diverge in the same direction (positive/negative infinity) then is it possible to rearrage the terms of one of the series to be the other series?

Example $a_n=\sum \limits_{n=1}^{\infty} \frac{1}{n}$ and $b_n=\sum \limits_{n=1}^{\infty} \frac{1}{\sqrt{n}}$

Now, I do understand that these are two different divergant sequences; however, it is possible to rearange the terms of the $b_n$ to equal the terms in the same order as $a_n$.

For $n = 1$ both have the same term. for $n=2$ $a_n = 1+ 1/2$ while $b_n$ at $n=2$ is $1/\sqrt{2}$ but there does exist (we do have an infinity number of terms to work with) some value of $b_n$ so that $b_2 + b_n = a_1+a_2$ and because we have an infinity number of terms to work with and the properties of addition, with the fact that both $a_n$ and $b_n$ have the same limiting sum, thus it should be possible to regroup the terms of $b_n$ to math the terms of $a_n$ thus $b_n$ and $a_n$ are the really the same sequence because of the properties of addition.

Thus only convergant series are unique? The motivation of this question is I just read about images of functions and mappings. Albeit, I didn't understand all the mechanices.

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I'm not sure what exactly you mean by two series being the same. For example, clearly the series $\sum_n 1/n$ and $\sum_n 1/\sqrt{n}$ are not the same, though they both diverge. –  Christopher A. Wong Mar 4 '13 at 0:52
    
@ChristopherA.Wong but you could rearrage the terms and group them together to make the same series, I'll edit my question to be more clear. –  yiyi Mar 4 '13 at 1:19
    
Perhaps you want your series to have terms converging to $0$, and you want to allow your "rearrangements" to have infinitely many terms, so if $\sum_n a_n$ and $\sum_n b_n$ are given, you want to start with a, perhaps infinite, subsequence of $a_i$s adding up to $b_1$, then another subsequence adding up to $b_2$, etc. Is this what you have in mind? –  Andres Caicedo Mar 4 '13 at 1:26
    
@AndresCaicedo 1. Yes, that is along the lines I was thinking. 2. Shouldn't it work if both series have the same limiting sum? –  yiyi Mar 4 '13 at 1:29
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Say one series is $4+ 1 +1/2 + 1/3 + 1/4+ \dots$ and the other is $1+1/2+1/3+1/4+\dots$. How do you rearrange the first one? (Where does the $4$ go?) –  Andres Caicedo Mar 4 '13 at 1:33

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up vote 2 down vote accepted

No, because the rearrangement operations you're describing are only valid on absolutely convergent series.

Consider, for instance, the series $$1+2-2+3+4-4+5+6-6+\ldots$$ consisting of all positive integers and all even negative integers. It clearly diverges to $\infty$. But I can rearrange the series as $$1-2+2-4+3-6+4-8+\ldots$$ which diverges to $-\infty$.

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Am I on the right track to think that there is a higher level mathematical operation possible? Is it possible to map two series together, I have seen it in the book I read using integrals as an example, thus it must be possible with series. Really, need to learn more. Thanks for your clear counter example. Nice work. –  yiyi Mar 4 '13 at 1:40
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Generally speaking, I don't think so. For example even with rearrangement and grouping you cannot turn $1+3+0+0+0+\ldots$ into $2+2+0+0+0+\ldots$. –  user7530 Mar 4 '13 at 2:34
    
Know of a good book on this topic? –  yiyi Mar 4 '13 at 5:33

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