Lesson 24a - Absolute Conditional Convergence

Lesson 24a - Absolute Conditional Convergence -...

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Absolute Convergence We say a series is absolutely convergent when it we look that the absolute values of the terms, and that new series converges: That is given: , then if the series: converges, then we say that it is absolutely convergent. 1 n n b 1 | | n n b Why should we consider the absolute series? That is summed up in the following theorem: Theorem: If a series is absolutely convergent, then the series itself converges. Note: This series talks nothing about divergence, only when the absolute series converges.
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Absolute Convergence Proof Say we have a series where converges. This means we have: or 1 n n b 1 | | n n b | | n n b b | | n n b b This means we know that | | | | n n n b b b Manipulating this inequality we get: l l if th t th b th i tt | | 2 | | 0 n n n b b b Clearly if converges, then so must , thus by the comparison test: converges as well. If we split this into two series: 1 | | n n b 1 | | 2 n n b S b b n n | | | | n n b b S but we know that the absolute series also converges (to say T) we get: n 1 1 1 n n which gives: making it converge as well. T b S n n 1 T S b n n 1
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What happens when it is not absolute?
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Lesson 24a - Absolute Conditional Convergence -...

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