Talk:Taylor series

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Multivariate Taylor Series[edit]

Why was the section on `multivariate Taylor series' removed by (Compare the version of 17:53, 2006-09-20 vs that of 17:55, 2006-09-20). I am going to add it again, unless someone provides a good reason not to. -- Pouya Tafti 14:32, 5 October 2006 (UTC)

I agree with Pouya as well! There's no separate article on multivariate Taylor series on wikipedia, so it should be mentioned here.Lavaka 22:22, 17 January 2007 (UTC)
I have recovered the section titled `Taylor series for several variables' from the edition of 2006-09-20, 17:53. Please check for possible inaccuracies. —Pouya D. Tafti 10:37, 14 March 2007 (UTC)

The notation used in the multivariate series, e.g. fxy is not defined. Ma-Ma-Max Headroom (talk) 08:46, 9 February 2008 (UTC)

Can someone please check that the formula given for the multivariate Taylor series is correct? It doesn't agree with the one given on the Wolfram Mathworld article. Specifically, should in the denominator of the righthand side of the first equation not be ? As an example, consider the Taylor series for centered around . As it is, the formula would imply that the Taylor series would be instead of . Note that the two-variable example given in this same section produces the second (correct, I believe) series, contradicting the general formula at the start of the section. Ben E. Whitney 19:14, 23 July 2015 (UTC)

It's correct in both. Using your function and the conventions of the article, we have
as required. Sławomir Biały (talk) 21:08, 23 July 2015 (UTC)
Oh, I see! I think I'd mentally added a factor for the different ways the mixed derivatives could be ordered without realizing it. Should have written it out. Thank you! Ben E. Whitney 15:56, 24 July 2015 (UTC)
No worries. This seems to be a perennial point of misunderstanding. It might be worthwhile trying to clarify this in the article. Sławomir Biały (talk) 16:02, 24 July 2015 (UTC)

Madhava of Sanfamagrama[edit]

Actually, I think Archimedes should be accredited with the first use of the Taylor series, since he used the same method as Madhava: using an infinite summation to achieve a finite trigonometric result. Liu Hui independently employed a similar method 400 years later, but still about 800 years prior to Madhava's work, although the Wikipedia article on Liu Hui does not reflect this.

In fact, it would have been quite easy for them to perform the same task as Madhava. It isn't difficult to square an arc (albeit in an infinite number of steps) using simple Euclidean geometry. I believe that Archimedes and later Liu Hui were aware of this. Last time I heard about it was at a History and Philosophy of Mathematics conference in 1998 at the Center for Philosphy of Science, University of Pittsburgh. Anyone care to dredge up a reference?

Taylor series with Lagrange and Peano remainders[edit]

Why there's nothing about those two remainders in the article?

Complete Sets[edit]

The Taylor series predates the ideas of complete basis sets, the loose ends of which were not fixed until 1905 with the square integrability condition. The TS is merely a statement of the completeness of the polynomial, where each term of the sum is regarded as an element of a complete (but not necessarily orthonormal) set. If a function f(x) is written as the sum of an orthonormal polynomial set, the nth derivative of f appearing in the TS simply extracts the nth coefficient of the orthonormal sum. (talk) 09:39, 27 June 2016 (UTC)

This is not true. The Weierstrass approximation theorem comes closest to what you are articulating here, which states that continuous functions on compact sets can be uniformly approximated by polynomials, but the polynomials need not be truncations of the Taylor series. Indeed, it is easy to construct examples of functions whose Taylor series does not converge to the function, although these functions will be approximated by other sequences of polynomials. The question of approximating in L^2 is qualitatively very different. There are families of orthogonal polynomials that give series expansions of functions, but in general there is no relationship between the series expansions that one gets in this way and the coefficients of the Taylor series. Sławomir
00:36, 28 June 2016 (UTC)

Bounding the error[edit]

I think "Taylor series" wikipedia page should have a section or link to another wikipedia page, explaining how to bound the error made by the a Taylor polynomial of degree n. — Preceding unsigned comment added by Mredigonda (talkcontribs) 14:08, 19 February 2017 (UTC)

That is the subject of a different article, Taylor's theorem. You have to follow the link to get to that from the relevant section(s) of this article. Sławomir Biały (talk) 14:25, 19 February 2017 (UTC)