What is difference between $$\sum_{n=1}^{k} f(n)+\sum_{n=1}^{k} g(n)$$ and $$\sum_{n=1}^{k} [f(n)+g(n)] $$
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Why would you expect any difference? – Simply Beautiful Art Jul 03 '17 at 19:57
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İs there no difference? – Soru Jul 03 '17 at 19:57
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@Soru There is no difference whatsoever, since addition is commutative and associative. – Franklin Pezzuti Dyer Jul 03 '17 at 20:01
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1No difference as long as the sum is finite. – sharding4 Jul 03 '17 at 20:01
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Not even a shred of beginning of a difference. – Bernard Jul 03 '17 at 20:03
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Thank you everyone. İf $k$ to infinity? – Soru Jul 03 '17 at 20:05
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Then we get silly things such as: $$0=\sum_{n=1}^\infty (1-1)\ne\sum_{n=1}^\infty1-\sum_{n=1}^\infty1=\infty-\infty$$ – Simply Beautiful Art Jul 03 '17 at 20:09
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If both are convergent series separately, then your statement holds – Joel Jul 03 '17 at 20:14
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I understood..Thank you so much. – Soru Jul 03 '17 at 20:18
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@Joel False. If both are absolutely convergent then the statement holds. – Simply Beautiful Art Jul 03 '17 at 23:49
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@SimplyBeautifulArt it holds from simple sum of limits theorems. Most of the time you need absolute convergence, but not here. – Joel Jul 03 '17 at 23:56
2 Answers
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If your $+$ operation is associative (which is the minimum required for operations to make any sense), then there is absolutely no difference.
Caligula
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There is no difference.
Note that if the summation was up to infinity, then there actually would be difference since you cannot change the order of the summation if the sum does not absolute converges. But in your case the sum is finite, therefore it is just a summation between a finite number of elements, and there is no problem in splitting up the sum.
Mickey
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