Definition talk:Abel Summation Method

The underlying definitions and/or approaches aren't fully documented, so the risk of handwavery/apparent-prestidigitation seems high.

The Abel Summation Method is regular, linear, and stable - these properties as they relate to this summation method should be proved somewhere - on this page or elsewhere? These three properties could potentially be discussed within the non-yet-existing summation method page, which is referenced in Abelian Theorem. Maybe summation method should include an outline of these three properties and a table of known summatation methods along with which of the three properties they obey?

Furthermore, the Abel Summation Method is consistent with but more powerful than the Cesàro Summation Method. Should this page even be a definition given that the Cesàro Summation Method is not currently a definition?

The page called Cesàro Mean probably needs to be renamed. I have never explored that. It was put together by someone as their first and only contribution. What we need is a) a page called Definition:Cesàro Mean which defines what it actually is (if there is such a thing), and then b) the page currently called Cesàro Mean needs to be renamed, and amended so as to invoke the definition, so then the reader understands what the thing is and what its properties are, and so on. And then c), for Definition:Cesàro Summation Method: if that is the same thing as Definition:Cesàro Mean this needs to be explained. If not, then that needs to be explained as well. Whatever is what.

Incidentally, it is paramount that contributors are not of the mindset "if you are on this page you ought to know what I'm talking about so I don't need to define my terms". The plan is for $\mathsf{Pr} \infty \mathsf{fWiki}$ to teach as well as document. If a page leaves someone thinking "I didn't understand that", we're doing it wrong. --prime mover (talk) 07:26, 8 May 2020 (EDT)

With regards to the "All and any advice as to how to implement this adequately is requested of anyone." comment - it might be interpreted as a request for how to implement the method to sum various example series. The following examples outline the approach used to apply the method and are not thorough - if they are to be used on the main page then they would need to be expanded:

$1 - 1 + 1 - 1 + \cdots$ sums to $\frac 1 2$ when you consider the limit of $1 - x + x^2 - x^3 + \cdots = \frac 1 {1 + x}$. Maybe this particular example, Grandi's series, deserves its own page?

Definition:Grandi's Series should indeed have its own page. We do have this: Divergent Sequence may be Bounded -- which uses Grandi's series as its canonical example, just that nobody has got around to linking it up with the man's name. It is invoked on Luigi Guido Grandi's page but that's as far as it went.

Definition:Grandi's Series has now got its own page. --prime mover (talk) 08:37, 9 May 2020 (EDT)

Oh, and incidentally, I have no patience with the approach that says "$1 - 1 + 1 - 1 + \cdots$ sums to $\frac 1 2$" -- it doesn't and it can't because it is not convergent, whatever Euler wrote about it. --prime mover (talk) 07:29, 8 May 2020 (EDT)

$1 − \frac 1 3 + \frac 1 5 − \frac 1 7 + \cdots$ sums to $\frac \pi 4$ when you consider the limit of $x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots$, which is the Power Series Expansion for Real Arctangent Function that is valid for the required limit.

I am happy to create the various pages mentioned above, and probably include many other summation methods; however, I have no wish to create a "mess" that others may feel requires a different approach - comments/advice are welcome.

NOTE: The second summation method mentioned on the page is the A*-method - should this be split into a second page or given that it's closely related should we keep it where it is? --John Coupe (talk) 06:54, 8 May 2020 (EDT)

This page should contain no more and no less than a definition of what an (or the) "Abel summation method" actually is. I have not been able to find a precise definition of what exactly it is. I have been led to believe that it defines a type of summation method, that is, such-and-such a summation can be defined as "of type Abel" if and only if (some property or properties). I may be wrong, and what is being defined here is "the" Abel summation method.

So, given that we can find (and document) exactly what an (or the) Abel summation method is, any pages which prove stuff about an (or "the") Abel summation method should be in their own pages, in a category called "Abel summation method" (or "methods", depending (as above) on whether it is "an" or "the"), and a library of such proofs can then be generated here.

This page arose purely as an attempt by me to work my way through the various dictionaries of mathematics that I have accumulated -- a far more challenging task than I imagined, considering the many gaps in coverage that we have on $\mathsf{Pr} \infty \mathsf{fWiki}$.

As for the $A*$-method: separate page. One of the philosophical approaches of $\mathsf{Pr} \infty \mathsf{fWiki}$ is to have one concept per page. (This is not completely rigorous, we haven't got round to tidying everything up yet, particularly the more complicated pages containing vast quantities of infodump from contributors espousing widely different philosophical approaches.) Enforcing the "one page one concept" rule takes away all the anguished decisions that need to be made "do I bracket this together with that? Do we split this page up?" Then if we want to gather pages together into one, we use the transclusion method, and that is super-easy to do when every concept is modular.

So, if you know your way around this area, fill your boots. Please note the house style -- it is considerably different from what is produced by most practitioners of $\LaTeX$, the main point of issue being that the source code is to be consistent and readable. The style that has evolved is optimal. --prime mover (talk) 07:17, 8 May 2020 (EDT)