How to prove Carrier's rule for divergent series?

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    Divergent Series
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Discussion Overview

The discussion revolves around Carrier's rule for divergent series, with participants exploring the nature of asymptotic series versus convergent series. The scope includes theoretical considerations and the implications of using different types of series for approximations.

Discussion Character

  • Debate/contested, Conceptual clarification

Main Points Raised

  • Some participants suggest that asymptotic series can provide good approximations with only a few terms, while convergent series may require many terms for similar accuracy.
  • Others argue that it is possible to have rapidly converging series and that not all asymptotic series are "good," challenging the notion that convergence inherently limits approximation quality.
  • A question is raised about the existence of a general theorem regarding the relationship between convergence and approximation quality.

Areas of Agreement / Disagreement

Participants express differing views on the effectiveness of asymptotic versus convergent series for approximations, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants highlight the need for careful consideration of the definitions and properties of series, as well as the conditions under which they are evaluated, but do not resolve these complexities.

Count Iblis
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http://en.wikipedia.org/wiki/George_F._Carrier" :confused:
 
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Did you see the word "humorous" in that description? There is no proof, its a joke.
 
HallsofIvy said:
Did you see the word "humorous" in that description? There is no proof, its a joke.

Yes, I know. But it is still the case that a few terms of an asymptotic series wil give you a good approximation while if you use a series that converges you typically need may terms to get a similar approximation.

This is then alleged to be caused by the fact that demanding convergence is an extra burden that comes at the expense of how good the approximation can be after only a few terms of the series.

Is there a general theorem about this?
 
Count Iblis said:
Yes, I know. But it is still the case that a few terms of an asymptotic series wil give you a good approximation while if you use a series that converges you typically need may terms to get a similar approximation.
No, that is not the case. It is quite possible to have a convergent series that converges rapidly or an asymptotic series that is very slow. It's just that there is no reason to talk about asymptotic series like that. We only look at "good" asymptotic series.

This is then alleged to be caused by the fact that demanding convergence is an extra burden that comes at the expense of how good the approximation can be after only a few terms of the series.

Is there a general theorem about this?
 

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