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In my opinion, I still believe that sum of two divergence series can be convergence series.

But here is my proof:

First, we have an obvious result: sum of two convergence series is convergence series.

Take opposite of this statement is: if exist one series is (non-convergence) so cannot be (non-convergence)

$==>$ if exist one series is divergence, so sum always be divergence.!!! (because non-convergence series is divergence series and vice verse)

If my proof is wrong, please tell me where and give me an example that sum of two divergence series can be convergence series please.

Thanks :)

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Take $\sum{1\over n}$ and $\sum{-1\over n}$ (there are less trivial examples). It's true if both series consist of nonnegative terms. –  David Mitra Jun 3 '12 at 16:58
    
@DavidMitra yes. I have thought your example before, but it's too special. –  hqt Jun 3 '12 at 17:00
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What do you actually mean by adding two series? Do you mean $$\left(\sum_{n=1}^{\infty} a_n \right) + \left(\sum_{n=1}^{\infty} b_n \right)$$ or $$\sum_{n=1}^{\infty} \left(a_n + b_n \right) ?$$ –  user17762 Jun 3 '12 at 17:00
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Too special for what exactly? –  anon Jun 3 '12 at 17:00
    
Take opposite of series and plus together is too special in my opinion. –  hqt Jun 3 '12 at 17:03
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2 Answers

up vote 2 down vote accepted

If $a_n+b_n = c_n$ and $\sum_n c_n$ is a convergent series, then $b_n=c_n-a_n$, so $\sum_n b_n$ differs from $\sum_n (-a_n)$ by the convergent series $\sum_n c_n$. In particular, $\sum_n b_n$ is convergent if and only if $\sum_n a_n$ is.

If you start with a particular divergent series $\sum_n a_n$, every series you can add termwise to $\sum_n a_n$ to obtain a convergent series has the form $\sum_n(-a_n+c_n)$, where $\sum_n c_n$ is whatever convergent series you like.

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The proof is wrong and here is why. The theorem states that if we add two convergent series then we get a new convergent series. So that statement is equivalent to the following. If a divergent series was produced by sum of two series then these two series were not convergent series (so at least one is divergent).This is different than saying adding two divergent series produces a divergent series. It is like having A implies B this is indeed equivalent to notB implies notA but is certainly different from notA implies notB.

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