Talk:Taylor series: Difference between revisions

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:::*:{{xt|1=In my opinion, if a statement is so obvious that it falls under BLUE, it doesn’t need to be explained to the reader and can be removed from the article.}} That also includes the shorter notation? [[User:Dedhert.Jr|Dedhert.Jr]] ([[User talk:Dedhert.Jr|talk]]) 03:36, 4 February 2026 (UTC)

:::::*{{re|Dedhert.Jr}} What do you mean by a shorter notation? Can you give an example from the article? [[User:Z1720|Z1720]] ([[User talk:Z1720|talk]]) 03:54, 4 February 2026 (UTC)

:::::*:When I wrote “shorter notation”, I meant it is basically to abbreviate the long writing of mathematical symbols with the shorter one, such as [[Taylor series#Taylor series in multiple variables]]. Shorter notation may be useful for saving time on writing long mathematical symbols, but if you want to write it in terms of many symbols, you have to write step-by-step with the given index (i.e., the list of numbers). Nevertheless, long writing symbols may be needed due to ~~introductory~~ of mathematical concepts. I would rather considered this as a [[WP:BLUE]], just like the way of writting long additions <math> 1 + 2 + \dots + n </math> into <math> \sum_{i=1}^n i </math>.

:::::*:When I wrote “shorter notation”, I meant it is basically to abbreviate the long writing of mathematical symbols with the shorter one, such as [[Taylor series#Taylor series in multiple variables]]. Shorter notation may be useful for saving time on writing long mathematical symbols, but if you want to write it in terms of many symbols, you have to write step-by-step with the given index (i.e., the list of numbers). Nevertheless, long writing symbols may be needed due to of mathematical concepts. I would rather considered this as a [[WP:BLUE]], just like the way of writting long additions <math> 1 + 2 + \dots + n </math> into <math> \sum_{i=1}^n i </math>.

:::::*:I have made an edit by merging into the previous lines, where the bottom explanatory text does not ~~neccesarily~~ need source. In other words, it’s basically a footnote. But I can use {{tl|efn}} if I want. [[User:Dedhert.Jr|Dedhert.Jr]] ([[User talk:Dedhert.Jr|talk]]) 04:31, 4 February 2026 (UTC)

:::::*:I have made an edit by merging into the previous lines, where the bottom explanatory text does not need source. In other words, it’s basically a footnote. But I can use {{tl|efn}} if I want. [[User:Dedhert.Jr|Dedhert.Jr]] ([[User talk:Dedhert.Jr|talk]]) 04:31, 4 February 2026 (UTC)

Taylor series has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.

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May 12, 2011	Good article nominee	Listed

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Please fix the “Examples” section to make sense. For example,
1st example is about the “MacLaurin series”. I’m here to learn about the Taylor series. I see MacLaurin mentioned in 2nd paragraph but I wasn’t sure what “about zero” means. Maybe if the example section had a Taylor series example…
4th line says “so the Taylor series…” What does it mean by “so”. How does that so obviously follow from the MacLaurin example? I can’t see the connection.
6th line says “By integrating the above Maclaurin series” What? You just called it a Taylor series. Why are you integrating it? That’s not even what I get when I integrate it.
And so on.
I would like to see some examples in the “examples” section.
Ywaz (talk) 00:43, 14 October 2021 (UTC)[reply]

There is no possible way to write an article that will make it understandable to a person who has decided not to try to read it. —JBL (talk) 01:14, 14 October 2021 (UTC)[reply]

No, I agree with Ywaz. This is yet another appallingly written mathematical article which seems to assume the reader already knows what the author is talking about. I likewise got no further than the “Examples” section before I was totally baffled.

212.159.76.165 (talk) 16:33, 14 June 2022 (UTC)[reply]

I have fixed the grammar of section § Examples (misuse of “for” instead of “to”, and comma after “so”). I have also added a short explanation to “so”. For the remainder of Ywaz’s complaints, nothing more can be done: “about 0” is not in the article, “Maclaurin series” is defined twice, in the lead and in section § Definition, and can thus be supposed to be known. About “appallingly written”, I would be happy if you could propose a better way to write this article. But, as for every technical article, a minimal background is required for understand it. Here, nobody can understand the subject without having learnt first what is a series and a derivative. D.Lazard (talk) 17:35, 14 June 2022 (UTC)[reply]

I agree the examples section does not flow very well and would be helped by more steps to show how it works for people trying to learn this, rather than just stating the results. Another problem with this section is that the part that starts with “By integrating the above Maclaurin series” seems to be not quite right. It is the integral of -1 times that series. (You don’t get -x by integrating 1). It would also be helped by referring to the series for (it should be) -1/(x-1) or 1/(x-1) by a name or other identifier (like a parenthesized formula number as is often done) so the reader doesn’t think it is referring somehow to the immediately preceding series. Skaphan (talk) 18:18, 10 February 2023 (UTC)[reply]

I am also having trouble understanding this section. Right at the beginning, it says:

The Taylor series of any polynomial is the polynomial itself.

The Maclaurin series of

⁠ 1 / 1 - x ⁠

is the geometric series

1+x+x^{2}+x^{3}+\cdots .

So, by substituting

x

for

1 - x

, the Taylor series of

⁠ 1 / x ⁠

a = 1

1-(x-1)+(x-1)^{2}-(x-1)^{3}+\cdots .

I am having trouble understanding everything after “So, by substituting…” Why? Why are we substituting x for 1-x? How does this follow from the first Maclaurin series? It seems to have jumped past some explanation of what is being discussed.
Wikinetman (talk) 18:46, 16 February 2024 (UTC)[reply]

At this time, the “examples” section says:

“So, by substituting $x$ for $1 - x$ , the Taylor series of $⁠ 1 / x ⁠$ at $a = 1$ is

$1-(x-1)+(x-1)^{2}-(x-1)^{3}+\cdots .$

By integrating the above Maclaurin series, we find the Maclaurin series of $ln(1 - x)$ , where…”

The example identifies $a = 1$ for the Taylor series, but then calls it a Maclaurin series, which should require $a = 0$ . In other words, “by integrating the above Maclaurin series” is not correct, the equation above is a Taylor series, not a Maclaurin series. Therefore, I suggest this be edited to:

“By integrating the above Taylor series, we find the Maclaurin series of $ln(1 - x)$ , where…”

I humbly present this concern recognizing I could be mistaken!
— Preceding unsigned comment added by Caireau (talk • contribs) 10:50, 16 July 2024 (UTC)[reply]

It’s a bit tricky perhaps because the variable name is being re-used. To clarify by using different variable names, the claim is that the “Maclaurin series” for the function ⁠

\textstyle u\mapsto \int {du/(1-u)}=\ln(1-u)

⁠ centered at ⁠

u=0

⁠ is the opposite of the “Taylor series” at ⁠

x=1

⁠ for the function ⁠

x\mapsto \ln x

⁠, under the change of variables ⁠

x=1-u

⁠. Aside: while introductory calculus textbooks nowadays tend to be very pedantic about a distinction between “Taylor series” and “Maclaurin series” in actual mathematical literature these are routinely conflated and given either name at the author’s whim, and it’s seldom if ever confusing in context. –jacobolus (t) 13:00, 16 July 2024 (UTC)[reply]

I follow. Thanks for that clarification.

I can see how the difference between Maclaurin and Taylor series can be pedantic since there’s nothing special about a=0. The prose of the explanation seems to have tripped me up in this case. Caireau (talk) 10:45, 17 July 2024 (UTC)[reply]

I think the section is definitely at least mildly confusing and could probably be copyedited/rewritten for clarity, if anyone wants to take a crack at it. –jacobolus (t) 15:41, 17 July 2024 (UTC)[reply]

The section “List of Maclaurin series of some common functions” says the Maclaurin for the Exponential function converges for all values of X. I disagree. When I first took calculus in the AP match in high school, I studied the Maclaurin series and learned it converges for values of X close to 0. In the Taylor Series, the value of X needs to be close to A. From what I understand, when computing the Exponential function, one takes the integer portion of the exponent and calculate the value by the Power function, or use an array to get the value, then plugged the decimal portion of the exponent into the Maclaurin series, and multiply the two results.

I set up an Access VBA stored procedure that calculates the value of 88.21389 of the Exponential function using Maclaurin series, subtract it from the value plugged into the EXP function, and the difference is “1.54269E+37”. Dkf12 (talk) 18:30, 26 September 2024 (UTC)[reply]

A general Taylor series only converges within a disk around its center, and in general the radius of that disk is set by wherever the closest singularity is. But the series for the exponential function in particular converges in the entire complex plane. I’m not sure what’s happening with your VBA script, but (a) in general I would not recommend implementing numerical routines in VBA, and (b) just because a series converges doesn’t mean it is an effective computation tool in practice. –jacobolus (t) 20:11, 26 September 2024 (UTC)[reply]

My VBA script is accurate as I described. Please refer to the website I found on Maclaurin series:

   https://brilliant.org/wiki/maclaurin-series/

It says quite clearly, “A Maclaurin series is a power series that allows one to calculate an approximation of a function f(x) for input values close to zero…”
As I noted, when I took calculus for AP math in high school, I learned the value of x of f(x) in a Maclaurin series needs to be close to 0. That makes perfect sense. To find values greater or less than 0, one needs to calculate the power or root of the integral portion, use Maclaurin for the decimal portion, and multiply the results. — Preceding unsigned comment added by Dkf12 (talk • contribs) 18:33, 6 March 2025 (UTC)[reply]

“close to zero” does not excludes “far to zero”. The Maclaurin series of the exponential function gives approximations of the function near to zero, but, for this specific function, one gets approximations for every ⁠

x

⁠, although they need more terms lor large ⁠

x

⁠. D.Lazard (talk) 19:59, 6 March 2025 (UTC)[reply]

The exponential function is an entire function meaning it can be expressed as a sum of a single power series, in this case

\textstyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}

for all

x\in \mathbb {R}

. See also radius of convergence of a power series, which you can calculate as

R=\left(\limsup _{n\to \infty }{\sqrt[{n}]{|c_{n}|}}\right)^{-1}=\left(\limsup _{n\to \infty }{\sqrt[{n}]{\frac {1}{n!}}}\right)^{-1}=+\infty

I don’t know how accurate your script is but when I plug the value 88.21389 into a simple C++ script (with floats and no optimization for rounding errors, just a simple sum from the first to the last) I get within 0.0001% of the correct result (2.04553e+38) after just 140 terms. So either you have a very big rounding error or some other problem (I am not even getting your difference for any partial sum). MartinVitVavrik (talk) 18:20, 7 March 2025 (UTC)[reply]

Try plugging different values into the formula for Mclaurin. I find that the result is far different from the value of EXP(X) when I plugged in 57 for X. Values less than 57 do work out. — Preceding unsigned comment added by Dkf12 (talk • contribs) 22:19, 9 March 2025 (UTC)[reply]

That’s a very good indication of exactly where your implementation of large integer arithmetic is being swamped by overflow or other rounding errors. The convergence of the power series for the exponential function is a very straightforward application of the ratio test, and this fact is covered in every standard calculus textbook (e.g., it’s Example 6.16 here). —JBL (talk) 22:52, 9 March 2025 (UTC)[reply]

It looks like the section “Examples” and “List of Maclaurin series …” have similar content, but the latter one lists many examples of series. The first one looks unsourced and could probably be removed. And I might have to agree with @Lapasotka in this oldid. Dedhert.Jr (talk) 03:55, 24 December 2025 (UTC)[reply]

@Jacobolus. Do you have any opinion of this? Dedhert.Jr (talk) 12:04, 1 February 2026 (UTC)[reply]

One section is apparently intended to be a couple of introductory examples, though I’m not sure how helpful they are to less technical readers. The other section is apparently intended to be a more significant reference list. I don’t think either section includes anything unverifiable, but I don’t think anyone would mind terribly if you remove the first section. There are probably better ways of providing initial motivation. –jacobolus (t) 18:28, 1 February 2026 (UTC)[reply]

It has been a while since this article has been reviewed, so I took a look and noticed lots of uncited statements, including entire sections. Dedhert.Jr made similar observations of some uncited sections above, so I’m pinging them. Should this article go to WP:GAR? I am happy to add citation needed templates if asked (and pinged), and use this script to help find those uncited statements. Z1720 (talk) 15:57, 31 January 2026 (UTC)[reply]

As a better alternative, you could just pick up any introductory calculus textbook to cite, and add the relevant footnotes yourself. Adding footnotes to uncontroversial basic material has some (frankly minimal) value, but adding eyesore templates to demand that someone else do the easy but somewhat tedious work to add them doesn’t really seem helpful. –jacobolus (t) 19:55, 31 January 2026 (UTC)[reply]

@Jacobolus: We are all volunteers, and posting concerns on a talk page is not demanding anyone to do anything: no one has to address the concerns. Unfortunately, I do not have the time, knowledge in this subject area, or interest to bring this article back to meeting the GA criteria. Z1720 (talk) 20:02, 31 January 2026 (UTC)[reply]

Sending things through the bureaucratic and antagonistic GAR process is a way of creating make-work for other volunteers, under a threat of taking away the green badge. The “concerns” here seem to me relatively trivial and largely unrelated to article quality: someone can stick a few footnotes on various WP:BLUE statements currently lacking them, but frankly that doesn’t make any significant substantive improvement to the article. From the examples I have seen GAR generally wastes a lot of time relative to the amount of improvement that results. If you don’t have any expertise to judge or improve the article per se, another idea would be to put some effort into recruiting subject experts, e.g. from a local college or by contacting people who have written about the topic. –jacobolus (t) 20:14, 31 January 2026 (UTC)[reply]

@Jacobolus: Citing information to verifiable sources is a major aspect of the GA criteria. WP:BLUE is an essay, which is not “Wikipedia policy, as it has not been reviewed by the community and may reflect various opinions.” Verifiability is a Wikipedia policy, and it states that prose, even if they seem obvious to some editors, does need to be cited (with some exceptions like WP:CALC, WP:PLOT and WP:LEAD). In my opinion, if a statement is so obvious that it falls under BLUE, it doesn’t need to be explained to the reader and can be removed from the article. Z1720 (talk) 20:30, 31 January 2026 (UTC)[reply]

By all means go pick up any introductory calculus textbook and start adding footnotes, since you consider that to be important. It won’t take you especially more effort than it would take anyone else. –jacobolus (t) 21:32, 31 January 2026 (UTC)[reply]

In my opinion, if a statement is so obvious that it falls under BLUE, it doesn’t need to be explained to the reader and can be removed from the article. That also includes the shorter notation? Dedhert.Jr (talk) 03:36, 4 February 2026 (UTC)[reply]

@Dedhert.Jr: What do you mean by a shorter notation? Can you give an example from the article? Z1720 (talk) 03:54, 4 February 2026 (UTC)[reply]

When I wrote “shorter notation”, I meant it is basically to abbreviate the long writing of mathematical symbols with the shorter one, such as Taylor series#Taylor series in multiple variables. Shorter notation may be useful for saving time on writing long mathematical symbols, but if you want to write it in terms of many symbols (you can call it “expansion”), you have to write step-by-step with the given index (i.e., the list of numbers). Nevertheless, long writing symbols may be needed due to the introduction of mathematical concepts. I would rather considered this as a WP:BLUE, just like the way of writting long additions $1+2+\dots +n$ into $\sum _{i=1}^{n}i$ .

I have made an edit by merging into the previous lines, where the bottom explanatory text does not necessarily need a source. In other words, it’s basically a footnote. But I can use {{efn}} if I want. Dedhert.Jr (talk) 04:31, 4 February 2026 (UTC)[reply]

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