Blanking on word for kind of convergence of a sum

1. Sep 19, 2013

Office_Shredder

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

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?

2. Sep 19, 2013

micromass

Staff Emeritus
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.

3. Sep 19, 2013

Office_Shredder

Staff Emeritus
That is exactly the term I was looking for. See, I knew the word principal was involved somehow :tongue:

4. Sep 21, 2013

jackmell

Fourier.

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