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Re: Conditionally convergent series
Posted:
Dec 11, 2010 11:04 AM
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I assume that" leaves all positive numbers of the series on their places" means that the positive numbers are in the same order as in the original series.
Make separate lists of the positive and negative numbers. Add from the positive numbers, in order, until the first time the sum is larger than t (Since, in order to be 'conditionally convergent' rather than 'absolutely convergent', the series of all positive terms must not converge, this will eventually happen). Now start adding numbers from the list of negative numbers until teh sum is back below t (again, the series of all negative terms must not converge so this will happen). Now go back to adding positive numbers until you are back above t. Since, in order to converge, the terms of the series are going to 0, we go "over" or "below" t by smaller and smaller amounts each time. That shows that the series converges to t.
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