Since elementary school, I was taught that addition is unaffected by order. Why does it matter in what order the terms of a conditionally converging series are added? Summation is addition, therefore shouldn't it have the same properties?
Why does order matter in a conditionally converging series?
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sequences-and-series
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0The rearrangement result shows that summation is not necessarily like addition. Finite sums and infinite "sums" do have some features in common. – 2012-11-03
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3Infinite operations are different from finite ones. Consider putting numbered balls into a bag and removing them again. If there are only a finite number of balls, it doesn't matter what order you put them in or take them out; the end result is the same. With an infinite number of balls, the order matters: if you put in 1 and take it out, put in 2 and take it out, and so on, the bag never contains more than 1 ball. But if you put in 1 and 2, take out 1, put in 3 and 4, take out 2, and so on, then even though every ball is eventually removed, the bag contains an ever-increasing number of balls. – 2012-11-03
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1Summation is not addition. Summing an infinite series involves both addition and limits, and limits are affected by order. – 2012-11-03