Talk:Taylor series

Potential Error in "Examples"

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)

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)

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)

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)

List of Maclaurin series of some common functions

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)

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)

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)

"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)

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)

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)

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)

Two similar sections

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)

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

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)

I'll add the Taylor series of a given function in each article. Dedhert.Jr (talk) 01:45, 7 February 2026 (UTC)

Article review

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)

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)

• @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)

  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)

• @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)

  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)

  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)

• @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)

  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)

  @Dedhert.Jr: I do not think shorter notations need citations because they would be covered under WP:CALC. Z1720 (talk) 01:04, 7 February 2026 (UTC)

  Got it. Dedhert.Jr (talk) 01:27, 7 February 2026 (UTC)

• Comment. It is mandatory in these processes to check the cited sources before tagging a statement. Rwo of the three tags added were already fully supported by actual citations. Consider this a warning not to omit this step. Sławomir Biały (talk) 08:48, 1 April 2026 (UTC)

  @Sławomir Biały: I add citation needed templates to statements missing an inline citation. In this statement you removed citation needed tags in a paragraph which do not have inline citations. The citation in the previous paragraph (ref 17) and after the math equation (ref 18) do not fulfil GA criteria 2b, which states that statements "must be cited no later than the end of the paragraph (or line if the content is not in prose)." Ref 17 and 18 are not in the same paragraph, so whichever inline citation is used to verify this information should be added at the end of the paragraph. Z1720 (talk) 15:55, 1 April 2026 (UTC)

  The citation is in that paragraph, and logically belongs in the first sentence. It clearly applies to the whole paragraph. In the future, please exercise more caution before adding spurious tags. If you are using a script to perform automated edits without checking citations, also be aware that sometimes displaymath in the middle of a logical paragraph will falsely register as a paragraph break. Please do not "correct" me again in this thread. Consider yourself warned to use more caution in the future. Sławomir Biały (talk) 16:04, 1 April 2026 (UTC)

  Sounds like the footnote should be moved from just before the equation to the end of the paragraph. It's obvious to most readers when the text says "the following theorem" and provides a footnote that the follow-up sentence about the content of the theorem is also supported by the source linked in the footnote. But this runs afoul of some technical trivia in the GA guidelines. User:Sławomir Biały, the general best practice is to assume that Wikipedia GA reviewers will be extremely pedantic and kind of dense. –jacobolus (t) 16:05, 1 April 2026 (UTC)

  No, the citations are in the right place; completely in accordance with the GA criteria, which are pretty permissive regarding the actual placement of citations. The equation is part of the paragraph and the citations appear at the beginning of the paragraph, before elaboration (from those same sources). Z1720 is just completely misrepresenting what the GA criteria require here (weirdly Z actualy says, correctly, that statements "must be cited no later than the end of the paragraph", without apparently understanding those words.) Sławomir Biały (talk) 16:09, 1 April 2026 (UTC)

  I think the general expectation is that citations must come after the text they support, but no later than the end of the paragraph. So then a footnote supporting a specific word can come after the word, a footnote supporting a specific sentence it can come after the sentence, and a footnote supporting the whole paragraph can come at the end of the paragraph. (In math articles we also have the convention that a footnote immediately before a block of "display" style math supports the mathematical expression.) –jacobolus (t) 16:18, 1 April 2026 (UTC)

  That's not a requirement, and in any case it is better here to leave the citation at the beginning of the block, because the content it supports is divided by a displaymath. But in any case, this was a case of an editor tagging for citation material that already satisfied the requirements of WP:V (the paragraph was cited), they just couldn't be bothered to check the source. That's just make-work for other editors. It's disruptive, and has nothing to do with good article criteria. Sławomir Biały (talk) 16:22, 1 April 2026 (UTC)

  I agree that editors who insist on such a change would do better to make it themselves (e.g. moving or duplicating the footnote to the end of the paragraph) rather than demanding that someone else to do so. –jacobolus (t) 16:39, 1 April 2026 (UTC)

  @Sławomir Biały: This review is comparing the article against the GA criteria, which typically requires citations after the information it is verifying (with some exceptions, like block quotes). If anyone thinks this situation is an exception, I suggest opening a thread at WT:GA to get more editor opinions. @Jacobolus: Although encouraged to make improvements, no editor has to edit an article. Adding a citation needed template, or stating how an article can be improved, is not "demanding that someone else to do so". We are all volunteers: if you do not want to make the necessary improvement, then don't. Z1720 (talk) 16:52, 1 April 2026 (UTC)

  Adding a "citation needed" template when there is already a citation provided for the given claim is not a helpful activity, but just a lazy cop-out which creates extra work for someone else for marginal if any benefit to readers. If you are reviewing an article and adding "citation needed" templates, you should spend the small amount of effort it takes to look at the immediately preceding footnotes to see if the sources given verify the claim you are concerned about. –jacobolus (t) 17:04, 1 April 2026 (UTC)

  @Jacobolus: In this case, there were no preceding inline citations earlier in the paragraph to check, because there were no citations in that paragraph. While that might take a small amount of effort for some, checking references and understanding what they say for mathematics articles is a large effort for editors like me, of which I am not interested in. It is up to editors who care about an article's status to ensure that the article meets the criteria of its status. If you think this interpretation of consensus is wrong, please open a discussion at WT:GA to open up this discussion. Z1720 (talk) 17:12, 1 April 2026 (UTC)

  Your first sentence is completely false, and you should retract it immediately. Here is the paragraph for which there "were no citations":

  The generalization of the Taylor series does converge to the value of the function itself for any bounded continuous function on (0, ∞), and this can be done by using the calculus of finite differences. Specifically, the following theorem, due to Einar Hille, that for any t > 0, $$\lim _{h\to 0^{+}}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}{\frac {\Delta _{h}^{n}f(a)}{h^{n}}}=f(a+t).$$ Here Δⁿ
  _h is the nth finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.

  Sławomir Biały (talk) 17:22, 1 April 2026 (UTC)

  @Sławomir Biały: Sorry, I was misremembering the paragraph with this diff, where the reference was removed with the citation needed template, making the paragraph uncited. In the situation where the citation needed template was added, the reference was in the middle of the paragraph. The expectation for me to check the source myself would require me to find a copy of the source (because there is no link to the source in the article) then understand the technical language in the article and source to verify the information. Community consensus states that this expectation is not required for reviewers to add a citation needed template, and that consensus will not change with this discussion (nor will my action of adding citation needed tags where I think they are required). If you would like it to be a requirement for reviewers, please open a thread at WT:GA. Z1720 (talk) 17:40, 1 April 2026 (UTC)

  (1) You state that you “misremembered” the paragraph, but you link the same diff that has just been quoted in full above, which clearly contains an inline citation in the sentence introducing the theorem. So this is not a memory issue. The citation is plainly present in the version you are referencing. (2) If you cannot check a source for its coverage, you probably shouldn't be challenging it in the first place. The Feller and Hille-Phillips sources are extremely standard, and you can find them in any well-stocked mathematical library. Sławomir Biały (talk) 18:03, 1 April 2026 (UTC)

  Well yes, if you want to review an article's sources, then you have to find the sources. That's always true. You can find copies at the Internet Archive: Feller, Hille & Phillips. If you really think there's a consensus that 'citation needed' reviewers should be as lazy as possible, then that seems like a serious problem. If you don't understand any of the technical content of the article or its sources, why are you trying to review whether the sources support the claims? You could instead pick an article to review that you have the prerequisite knowledge to make sense of. –jacobolus (t) 17:50, 1 April 2026 (UTC)

Two of the citation needed templates that I added were removed in these edits without accompanying refs. The cn tags went with the following statements (with an explanation following each statements):

• "Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments x." In the first sentence, I think saying that the series listed are "important" is an opinionated statement, which WP:WIKIVOICE says should be avoided. I think a citation is needed for that qualifyer to be used. In the second sentence, I think the same reasoning can be used for "valid": this is a qualifyer that should be cited instead of stated as fact in wikivoice.
• "The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series." I think a citation is needed to verify that the series is formally similar to the Newton series: I'm not sure in what way it can be "formally" similar, and if this is a technical term it should be explained or replaced. Furthermore, how it is similar is not explained in the article, so WP:CALC is not applicable. Instead of a ref, a solution might be to remove the phrase as off-topic for this article.

Opinions on resolving these can be posted below. Z1720 (talk) 18:52, 1 April 2026 (UTC)

Have you ever studied high-school level calculus? Saying that the Taylor series for the exponential function, natural logarithm, trigonometric functions, etc. are "important" is something like saying "Water is an important molecule" in the context of an article about biology; i.e. this is a WP:BLUESKY situation, and it's absurd to throw in a "citation needed" tag. Feel free to go track down a source if you really want (it should not be hard for you), but the template there does a massive disservice to readers and is completely unnecessary. –jacobolus (t) 19:32, 1 April 2026 (UTC)

• @Jacobolus: WP:BLUESKY (and WP:NOTBLUESKY, which says the opposite) are essays, where it "contains the advice or opinions of one or more Wikipedia contributors. This page is not an encyclopedia article or a Wikipedia policy, as it has not been reviewed by the community." However, WP:NPOV is a policy, which "describes a widely accepted standard that all editors should normally follow." I do not think this is a WP:IAR situation, and thus I think these statements will need citations. Z1720 (talk) 19:45, 1 April 2026 (UTC)

  Your claim that it is non-neutral to describe the most common Taylor series taught in every first-year textbook as "important" is completely absurd. Indeed, if you just go pick any Calculus book off the shelf, turn to the relevant section, and read it, you will find these series described as "important", "fundamental", "essential", "basic", "common", etc. If you really want, go find whichever introductory calculus book you want (or look for a "mathematics for engineers" or whatever book if you prefer) and add such a footnote, but it's not really going to be helpful to readers and it's a waste of other people's time and attention to demand that someone else do so. We're not ignoring rules, this is basic common sense. –jacobolus (t) 19:50, 1 April 2026 (UTC)

• @Jacobolus: No one, including you, has to add the citation as we are all volunteers. If you think I am demanding your time, you should open a thread about it at WT:GA or WP:ANI as I am contravening WP:NOTCOMPULSORY. Adding a citation needed or opening a discussion about potential uncited statements is not, in my opinion, demanding your time. Z1720 (talk) 20:06, 1 April 2026 (UTC)

  The citation needed template is superfluous and net harmful (it pokes the reader in the eye with a stick for absolutely no reason), and the person insisting on a footnote should go do the work of making the footnote. Or we can leave the status quo, which is entirely fine. Or feel free to substitute e.g. "standard" for "important", if you (in my opinion wrongly) believe that the latter is editorializing. –jacobolus (t) 20:20, 1 April 2026 (UTC)

  I have trimmed POV language from math articles plenty of times (e.g., removing "it is easy to see" and phrasing like that), but I don't have a real problem with "important" here. The section would read oddly without a brief summary at the beginning, "Several Maclaurin series expansions" sounds weird and arbitrary without a further qualifying adjective, and these are functions whose expansions get called "important" all over the place, like here for the exponential, natural logarithm and binomial series. I'm not hurt by "important"; I also wouldn't in principle object to a different adjective if one were suggested ("standard" sounds OK but a little drab). All told, it doesn't seem worth fussing over too much. Stepwise Continuous Dysfunction (talk) 23:01, 1 April 2026 (UTC)

References

Taylor series with multi variables

Why not just add f(T) = f(T_0) + 1/1! df(T_0) + 1/2! d^2 f(T_0) + 1/3! d^3 (T_0) + ... for calculating Taylor series but have all that complexness ~2026-90872-6 (talk) 12:43, 10 February 2026 (UTC)