Does the following relation holds for infinite sums?

Hey i am only wondering if it is possible that the equation below holds:

$$\sum\limits_{i=0}^{\infty} a_i(b_i - c_i) = \sum\limits_{i=0}^{\infty} (a_ib_i - a_ic_i) =\sum\limits_{i=0}^{\infty} a_ib_i - \sum\limits_{i=0}^{\infty} a_ic_i$$

Thanks!

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You appear to have a typo: I think that you mean $$\sum_{i=0}^\infty a_i(b_i-c_i)=\sum_{i=0}^\infty(a_ib_i-a_ic_i)=\sum_{i=0}^\infty a_ib_i-\sum_{i=0}^\infty a_ic_i\;.$$ Yes, provided that the three series all converge. –  Brian M. Scott Mar 18 '13 at 20:29
It was a typo yes. I fixed it. –  71GA Mar 18 '13 at 20:36
No, you haven't. $a(b-c)=ab-ac$ and not $ab-bc$. –  Asaf Karagila Mar 18 '13 at 20:37
Should still be $a_ic_i$ not $b_ic_i$. –  Thomas Andrews Mar 18 '13 at 20:37
I did fix that thanks! –  71GA Mar 19 '13 at 6:16

Recall the definition of a series is: $$\sum_{i=1}^\infty a_i=\lim_{n\to\infty}(a_1+\ldots+a_n).$$

The answer to your question follows from the fact that: $$\lim_{n\to\infty}a_n(b_n-c_n)=\lim_{n\to\infty}a_n b_n-\lim_{n\to\infty}a_n c_n,$$ whenever all the limits have a finite value.

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@GitGud: Finite in the sense that the actual limit of the series is finite. Not the index set. –  Asaf Karagila Mar 18 '13 at 20:31
A small misunderstanding on my part. Sorry. –  Git Gud Mar 18 '13 at 20:32
@GitGud: You should have left the comment, it was a good remark that made the answer slightly better understood. But I suppose a small edit would be even better. –  Asaf Karagila Mar 18 '13 at 20:36

It holds when $$\sum_{i=0}^\infty a_i b_i$$ and $$\sum_{i=0}^\infty b_i c_i$$ ar both unconditional convergent.

Or $$\sum_{i=0}^\infty a_i (b_i - c_i)$$ is unconditional convergent.

This is in the finite dimensional the case when they are absolute convergent, that means when $$\sum_{i=0}^\infty |a_i b_i|$$ is convergent.

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Do you really need absolute convergence? –  Asaf Karagila Mar 18 '13 at 20:28
Do you have an example for when this fails for a conditionally convergent series? –  Asaf Karagila Mar 18 '13 at 20:30
Interesting. I thought that $0=0$. –  Asaf Karagila Mar 18 '13 at 20:33
but $\infty -\infty$ is not zero, it is indeterminate –  Dominic Michaelis Mar 18 '13 at 20:33
But then $\sum a_ib_i=\infty$. This is not about conditional convergence anymore. It's just not a convergent series. –  Asaf Karagila Mar 18 '13 at 20:35