Talk:Power series

Explanation of Symbols Needed.

There is an uneven distribution of levels of understanding on this (and it seems, any mathematics related) page.

The problem is that many of the symbols are not explained. There is a lot that is easy to understand on this page, and the hypothetical you, the reader, will be going along fine "yeah yeah, uhuh, I get that," until all of a sudden, "What the hell is that??!"

And if you don't know, then you just don't know.

I would go through and fix this problem, But I somehow managed to get to Multivariable Calculus without seeing some of these symbols.

DoomedToBeTeaching (talk) 05:46, 8 March 2009 (UTC)

Which ones? 67.198.37.16 (talk) 17:37, 5 July 2016 (UTC)

Multiplication and division

I don't believe the division formula is correct.

Cheers

   Kris  — Preceding unsigned comment added by Krzysztofandrzejkrakowski (talk • contribs) 02:14, 9 August 2016 (UTC) 

Uniqueness of coefficients

Probably easy, but if f and g are two power series, and f=g for all x, then a_i=b_i for all i? Darcourse (talk) 20:39, 23 September 2019 (UTC)

What specifically in the article are you referring to?—Anita5192 (talk) 20:45, 23 September 2019 (UTC)

More like it's lack of presence in the article :) Darcourse (talk) 20:47, 23 September 2019 (UTC)

I think you can prove it with differentiation. f(0)=g(0) -> a_0=b_0, so these can be removed, and f_1=g_1. The differentials must also be the same, so f_1'(0)=g_1'(0) -> a_1=b_1, etc... Darcourse (talk) 20:53, 23 September 2019 (UTC)

For a formal series it is the definition of equality. For non-formal series (e.g., Taylor series), it is only true if you add that the series converges in some open neighborhood: for example, the series $\sum n!x^{n}$ and $\sum n^{n}x^{n}$ agree at every point and which they both converge. I don't think that suggests something that should be added to the article. --JBL (talk) 21:15, 23 September 2019 (UTC)

Misleading use of the word convergent

Mathematicians often use misleading grammar (unnecessarily confusing wording) when explaining themselves. The wikipedia pages on Power Series and Taylor Series offer a good example of unnecessarily confusing wording wherein the word "convergent" is misused grammatically. A complete, clear, understandable use of the word "convergent" identifies TWO things that move towards each other. TWO things must be identified. The Radius of Convergence section of the Power Series page now says "A power series will converge for some values of the variable x and may diverge for others", which completely ignores to specify WHAT THING the power series is converging towards! A power series can converge to a) a finite number, b) an analytical function, c) zero, d) infinity, but this wikipedia article talks about "convergent" as if the reader pre-knows that the power series is supposed to converge towards -- which is common parlance for mathematicians but is not obvious for the non-expert reader. Thus, for the sake of clarity and readability, this Power Series wikipedia explanation needs a small edit to specify WHAT THING the power series "will converge" towards. I tried to make this simple improvement "A power series will converge towards an analytic function for some values of the variable x and may diverge for others" but it was undone because people want to discuss it first, e.g. D.Lazard. So let's discuss. --Damon Turney (talk) 23:47, 8 December 2020 (UTC)

The sentence "The series converges" is standard, correct mathematical English. Anyone who has taken a first course in calculus will have been introduced to it, and no one who has not taken a first course in calculus could possibly understand any of what you are talking about.
Separately, you reverted my edit, but your edit summary (and the fact that you are the one who originally made the change) suggests you have no substantive objection to it. If that is correct, I ask that you self-revert. --JBL (talk) 01:20, 9 December 2020 (UTC)

I agree with you that it doesn't need to converge to an analytical function, so for that, I thank you. But let's be clear, we should allow non-experts to have a chance at understanding this Wikipedia article -- by using correct English. It is a simple thing to understand that the word "converge" must specify TWO things that are converging towards each other. A power series is just one thing, so we must specify what SECOND THING the power series could be "converging" towards. This whole issue of confusion is nicely and clearly explained in the convergent series article where it is clearly explained that "convergent" means that the series is converging to a non-infinite limit. Can we please insert a convergent series link into the first sentence of the Radius of Convergence subsection of the Power Series article? It would clear everything up. I will go ahead and insert convergent series. --Damon Turney (talk) 17:46, 9 December 2020 (UTC)

To editor Damonturney: It is standard that many English words have a mathematical meaning that differs from their common English meaning. These different meanings lead often to different grammatical syntax. For example, "unknown", "variable", and "indeterminate" are nouns in mathematics, while they are only adjectives in current English (also, "variables" are often things that do not vary, and "indeterminates" are well defined objects). In mathematics, the verb "to converge" means "being convergent", and therefore does not require to have any complement (for the definition of "being convergent", see Convergent series). Also, the convergence of a series does not requires that the series represents a function. For exemple, the fact that the series $\textstyle \sum _{n=0}^{\infty }1/n!$ converges is a theorem that can be proved without knowing that its sum (not its limit) is commonly called e. D.Lazard (talk) 10:24, 9 December 2020 (UTC)

No, incorrect, the words "unknown" and "variable" can also be NOUNS in English. Just look in a dictionary. Regarding "being convergent", you are also incorrect: "being convergent" also requires explanation of what SECOND THING it is "being convergent" towards. Regarding the series $\textstyle \sum _{n=0}^{\infty }1/n!$, I agree with you that it doesn't need to converge towards a function, and for pointing that out, I thank you.--Damon Turney (talk) 17:46, 9 December 2020 (UTC)

If we're doing the "just look in a dictionary" test, here is what Merriam-Webster has to say about "converge": "converge (intransitive verb) ... 3 : to approach a limit as the number of terms increases without limit; 'the series converges' " --JBL (talk) 17:54, 9 December 2020 (UTC)

Yes good point JBL, but the problem is that most people don't use that definition of "converge", or have it memorized, and most readers on Wikipedia are reading Wikipedia because they are trying to LEARN mathematics -- they are not already experts in mathematics, otherwise they would have no need for reading Wikipedia. Clarifying this point of confusion helps Wikipedia serve non-experts.--Damon Turney (talk) 18:07, 9 December 2020 (UTC)

I have reverted your change from "point of convergence" to "convergent point" since "convergent point" is a phrase that does not has any mathematical meaning, and is thus your own WP:original research. D.Lazard (talk) 11:05, 9 December 2020 (UTC)

Weirdly, that phrase was already present before Damonturney started editing. (In fact, it's been in the article a really long time. Damonturney removed it, but then later reinstated it ¯\_(ツ)_/¯.) --JBL (talk) 12:53, 9 December 2020 (UTC)

JayBeeEll, I reinstated "convergent point" because D.Lazard eliminated it initially and called for a "talk" session, so I was just giving D.Lazard a chance to talk about and edit everything before final changes were accepted.--Damon Turney (talk) 17:46, 9 December 2020 (UTC)

I understand why you did it; what I was trying to point out is that there is no need to discuss things that everyone agrees about, ergo there was no point in reverting something when you agree it is an improvement. (It's a moot point now -- we all have agreed that we agree, and it has been put back correctly.) --JBL (talk) 17:58, 9 December 2020 (UTC)

Great, thanks, sorry for the back and forth.--Damon Turney (talk) 18:00, 9 December 2020 (UTC)

Power series = function series

User:JayBeeEll You seem to disagree that a power series is a series of functions, despite the fact that I provided a reference for this. Could you explain your motivations?

Besides, I am generally against the use of intuitive, non-mathematical vocabulary like "infinite series". As far as I know, a series or a sequence is always "infinite". The emphasis ought to be on whether it is a series of functions or a numerical series. --L'âne onyme (talk) 18:02, 27 October 2021 (UTC)

Power series are certainly not always series of functions; for example over a finite field where polynomials are not in one-to-one correspondence with polynomial functions. As I said elsewhere, your personal idiosyncrasies are interesting but are not an acceptable basis for writing Wikipedia articles. --JBL (talk) 20:26, 27 October 2021 (UTC)

Ok, I didn't know that. So if I understand you right, a power series is actually a series of polynomials rather than functions ? Then I think it should be stated more clearly. In the whole article I didn't find anywhere a precise definition of a power series. Can we at least agree that "infinite series" is far too vague ?

Besides, as I also said elsewhere, you are bound to respect the rules of politeness on Wikipedia, especially since despite my personal "idiosyncracies" I have never broken the rules or engaged in edit warring on my side (which is not my intention).--L'âne onyme (talk) 21:15, 27 October 2021 (UTC)

Can we at least agree that "infinite series" is far too vague ? No, for reasons I have articulated elsewhere. --JBL (talk) 21:21, 27 October 2021 (UTC)

Can we at least agree that "infinite series" is far too vague ? No, as infinite series links to a definition.

... whether it [a power series] is a series of functions or a numerical series. It may be both, and also a series of monomials (formal power series). The distinction is exactly as relevant as deciding whether $x^{2}$ is a function (formally, it is not), an expression, a monomial, a polynomial, a quadratic form, or even a number, if a value has been assigned to x. D.Lazard (talk) 09:51, 28 October 2021 (UTC)