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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.
- Summer '00
- Calculus, SmartPen Lectures, Lecture Notes, MATH1004, Kyle Harvey