Blanking on word for kind of convergence of a sum

[Office_Shredder]

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?

 

[micromass]

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.

 

[Office_Shredder]

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 :tongue:

 

[jackmell]

Does anybody know what the word I am looking for is?

Fourier.

 

[CellCoree]

The word you are looking for is "Cesàro" convergence. This type of convergence, named after the Italian mathematician Ernesto Cesàro, is a form of summability where the limit of the partial sums of a series is considered instead of the series itself. This allows for a broader range of series to be considered convergent compared to the traditional definition of convergence. Other summability methods include Abel, Borel, and Euler summation.