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Originally Posted by phukraut
An example of molloby's idea appears on page 77 of Goldberg's Methods of Real Analysis (second edition):
[snip]
The book goes on to give the following theorem:
Let L be a conditionally convergent series of real numbers. Then for any real number x, there is a rearrangement of L which converges to x. However, if the series is absolutely convergent, then all rearrangements sum to the same value.
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Quote:
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Originally Posted by knifemissile
Thus, a conditionally convergent series can be reordered to sum to any value.
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Looks like you've been beaten to it!
It's an interesting concept though - but I guess intuitions can be appeased with the idea that you're essentially asking for the difference between two infinite (divergent) series. Their values being undefined it seems reasonable that depending on exactly how you ask the question you can get any answer you like.