Lesson 24a - Absolute Conditional Convergence

Thismeanswehave n 1 n 1 bn bn or bn bn

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Unformatted text preview: Convergence Proof | bn | bn Say we have a series where converges. This means we have: n 1 n 1 bn | bn | or bn | bn | This means we know that | bn | bn | bn | Manipulating this inequality we get: 0 bn | bn | 2 | bn | | bn | 2 | bn | Clearly if converges, then so must , thus by the comparison test: n 1 b | b n 1 n n n 1 n 1 n 1 | S converges as well. If we split this into two series:S bn | bn | but we know that the absolute series also converges (to say T) we get: n 1 n 1 S bn T which gives: making it converge as well. bn S T What happens when it is not absolute? We call a series that does...
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This note was uploaded on 02/07/2014 for the course MATH 1004 taught by Professor Mark during the Summer '00 term at Carleton CA.

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