Blanking on word for kind of convergence of a sum

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    Convergence Sum
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Discussion Overview

The discussion revolves around the concept of convergence for a specific type of sum, particularly focusing on the limit of a series as it approaches infinity. Participants explore terminology related to this type of convergence, specifically seeking a term that describes the limit of the sum from negative to positive infinity.

Discussion Character

  • Exploratory, Conceptual clarification

Main Points Raised

  • One participant presents a sum and expresses a desire to discuss its limit rather than its conventional convergence, seeking a specific term related to this concept.
  • Another participant suggests that the term being sought is the (Cauchy) principal value of the series, noting its relevance in both integrals and series.
  • A later reply confirms that the suggested term is indeed what the original poster was looking for, indicating that the term "principal" is involved.
  • Another participant introduces the term "Fourier" without further context or explanation.

Areas of Agreement / Disagreement

There appears to be agreement on the term "Cauchy principal value" as a suitable descriptor for the limit of the sum, while the mention of "Fourier" introduces an additional term that is not elaborated upon, leaving its relevance unclear.

Contextual Notes

The discussion does not clarify the specific conditions under which the principal value is applicable or how it relates to the concept of Fourier, leaving some assumptions and definitions unresolved.

Office_Shredder
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I have a sum
[tex]\sum_{n=-\infty}^{\infty} f(n)[/tex]
which I do not want to consider the convergence of in the normal sense, but I want to talk about the limit
[tex]\lim_{N\to \infty} \sum_{n=-N}^{N} f(n).[/tex]

I know that when this limit exists the sum is _____ convergent, or is a _____ sum, where _____ is something like principal, or first order, or perhaps a name like a Dirichlet sum (I'm making these up of course). Does anybody know what the word I am looking for is?
 
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Seems like you want the (Cauchy) principal value of the series. This is a well-known thing for integrals, but I've seen the term used for series too.
 
micromass said:
Seems like you want the (Cauchy) principal value of the series. This is a well-known thing for integrals, but I've seen the term used for series too.

That is exactly the term I was looking for. See, I knew the word principal was involved somehow :-p
 
Office_Shredder said:
Does anybody know what the word I am looking for is?

Fourier.
 

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