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11128418 No.11128418 [Reply] [Original]
Quoted By: >>11128427

Why is it with an infinite series you're allowed to arbitrarily rearrange and group up terms to sum them up into a whatever convergence point you want? Is this just a logical technicality? Because it seems to be so arbitrary that it's inapplicable this way.

>> No.11128427
Quoted By: >>11128451

>Why is it with an infinite series you're allowed to arbitrarily rearrange and group up terms to sum them up into a whatever convergence point you want?
you arent. is that what you meant to say?
you arent allowed to because if you have an infinite number of terms that add up to positive infinity, and an infinity number of terms that sum to negative infinity, you can have the sum converge at whatever value you want because you will always have an infinite number of remaining positive and negative terms

>> No.11128451

Let me see if I get this straight, are we talking in a case of an alternating series that results in something like infinity minus infinity, that because you are comparing two infinite sums the end convergence COULD be any arbitrary number, but not exclusively as a single solution?
No matter how many terms it takes you to get the constant convergence you desire (say 4 positive terms to 1 negative term as a ratio), the series extends infinitely and thus even if you're using more + than - terms to reach your definition because the series extends infinitely, you never have to face running out of terms within this skewed ratio?
But doesn't that reinforce the claim I'm asking about, that you indeed are allowed to rearrange and group terms, or is this (as asked above) specific to this type of series only?

>> No.11128617

First note that you can apply this technique only to so-called conditionally convergent series, i.e. series that are not absolutely convergent. If a series if absolutely convergent, then any permutation on the order of summands still converges, and to the same limit.

If you're startled by the length of any rigorous proof to Riemann's rearrangement theorem, there's no need to be - it all boils down to this key concept: One can prove that if a series is conditionally convergent, then the sum of all its positive terms tends to infinity, and the sum of all its negative terms tends to infinity. Suppose you have two infinite bags of such terms (the positive and the negative terms respectively of some conditionally convergent series), and you wish to find a permutation that makes your series converge to some desired limit . Well, start adding up positive terms until you've surpassed by a little bit; then stop, and start adding negative terms until you're below by a little bit; and so forth. When taking the limit of this process it's not hard (although a bit technical) to verify that the sequence of partial sums stabilizes around , converging to it as a limit.

It's also possible to make your series diverge to pos/neg infinity this way, and also to make it diverge altogether (by letting it oscillate between 1 and -1 for example).

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