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| − | Power series always converge absolutely for <math>x</math> in <math>|x-a|<R</math> where <math>R</math> is the radius of convergence. They always diverge for <math>x</math> in <math>|x-a|>R</math>. At points <math>x</math> where <math>|x-a|=R</math>, anything might happen. They might converge absolutely, or conditionally, or they might diverge. --[[User:Bell|Bell]] 20:56, 8 November 2008 (UTC) | + | Power series always converge absolutely for <math>x</math> in <math>|x-a|<R</math> where <math>R</math> is the radius of convergence. They always diverge for <math>x</math> in <math>|x-a|>R</math>. At points <math>x</math> where <math>|x-a|=R</math>, anything might happen. They might converge absolutely, or conditionally, or they might diverge -- and the left endpoint might do something different from the right endpoint. --[[User:Bell|Steve Bell]] 20:56, 8 November 2008 (UTC) |
Latest revision as of 15:57, 8 November 2008
Absolute/Conditional Convergence for Power Series
Ok, I've think I've figured out what this means, but if someone can contribute, I would appreciate it. It looks like conditional convergence for power series roughly refers to those endpoints that the ratio test fails to determine. If you use some other test, and find it converges, then it is either absolutely or conditionally convergent. If endpoints that cause it to be a positive (absolute) series, then they are part of the absolutely convergent interval. Only when the endpoint converges and it causes the series to alternate, while its absolute value fails, can you say that it is conditionally convergent.
I know, that was a terrible way to say that. Hopefully you can see what I mean; its easier to show that to describe.
Power series always converge absolutely for $ x $ in $ |x-a|<R $ where $ R $ is the radius of convergence. They always diverge for $ x $ in $ |x-a|>R $. At points $ x $ where $ |x-a|=R $, anything might happen. They might converge absolutely, or conditionally, or they might diverge -- and the left endpoint might do something different from the right endpoint. --Steve Bell 20:56, 8 November 2008 (UTC)
