Questions tagged [taylor-expansion]

Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

The Taylor expansion is the power series expansion of a function at a point. It represents a function as an infinite sum with terms calculated from the function's derivatives at that point. More precisely, It is defined as $$ \sum^{\infty}_{n=0}\frac{f^{(n)}(a)}{n!}(x-a)^n=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots $$

It happens often in applications that the Taylor expansion of $f$ at $a$ converges to $f$ (pointwise and locally uniformly) on some neighborhood of $a$: when this happens, the function is said to be analytic at $a$.

Applications:

A Taylor series is an idea used in computer science, calculus, and other kinds of higher-level mathematics. It is a series that is used to create an estimate (guess) of what a function looks like. Taylor Series are also used in power flow analysis of electrical power systems (Newton-Raphson method). Multivariate Taylor series is used in different optimization techniques; that is, you approximate your function as a series of linear or quadratic forms, and then successively iterate on them to find the optimal value.

References:

https://en.wikipedia.org/wiki/Taylor_series

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Geometric representation of Euler-Maclaurin Summation Formula

In Tom Apostol's expository article (here's a free link), upon seeing the figure below (or this from the Wolfram project) I was expecting more diagrams to come to continue the error decomposition of the shaded regions in the shape of "curved…
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How local is the information of a derivative?

I have read it a thousand times: "you only need local information to compute derivatives." To be more precise: when you take a derivative, in say point $a$, what you are essentially doing is taking a limit, so you only need to look at the open…
Michael Angelo
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Is it impossible to perfectly fit a polynomial to a trigonometric function on a closed interval?

On a closed interval (e.g. $[-\pi, \pi]$), $\cos{x}$ has finitely many zeros. Thus I wonder if we could fit a finite degree polynomial $p:\mathbb{R} \to \mathbb{R}$ perfectly to $\cos{x}$ on a closed interval such as $[-\pi, \pi]$. The Taylor series…
jskattt797
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An intuitive explanation of the Taylor expansion

Could you provide a geometric explanation of the Taylor series expansion?
Jichao
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Are Taylor series and power series the same "thing"?

I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general concept? How does either/all of these ideas relate to…
gone
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Could these polynomials be identified?

Looking a good approximation of the $n^{th}$ positive root of the equation $$\color{blue}{\tan(x)=k x}$$ As already done many times, I expanded as Taylor series around $x=(2n+1)\frac \pi 2$ and used series reversion. As a result, this…
Claude Leibovici
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When do Taylor series provide a perfect approximation?

To my understanding, the Taylor series is a type of power series that provides an approximation of a function at some particular point $x=a$. But under what circumstances is this approximation perfect, and under what circumstances is it "off" even…
user525966
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What is the difference between the Taylor and Maclaurin series?

What is the difference between the Taylor and the Maclaurin series? Is the series representing sine the same both ways? Can someone describe an example for both?
smaude
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Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$

Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$? The Taylor series of $f(x)$ at $0$ is $\sum_{n=0}^{\infty}…
Valtteri
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What kind of functions cannot be described by the Taylor series? Why is this?

It's true that I'm not familiar with too many exotic functions, but I don't understand why there exist functions that cannot be described by a Taylor series? What makes it okay to describe any particular functions with such a series? Is there any…
smaude
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On the vector spaces of Taylor Series and Fourier Series

Taylor series expansion of function, $f$, is a vector in the vector space with basis: $\{(x-a)^0, (x-a)^1, (x-a)^3, \ldots, (x-a)^n, \ldots\}$. This vector space has a countably infinite dimension. When $f$ is expressed as linear combination of the…
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What does it mean for a polynomial to be the 'best' approximation of a function around a point?

I think I understand how to use Taylor polynomials to approximate sine. For instance, if $$ \sin x \approx ax^2+bx+c $$ and we want the approximation to be particularly accurate when $x$ is close to $0$, then we could adopt the following approach.…
Joe
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Doubt about Taylor series: do successive derivatives on a point determine the whole function?

I'm currently relearning Taylor series and yersterday I thought about something that left me puzzled. As far as I understand, whenever you take the Taylor series of any function $f(x)$ around a point $x = a$, the function is exactly equal to its…
David O.
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Do the Taylor series of $\sin x$ and $\cos x$ depend on the identity $\sin^2 x + \cos^2 x =1$?

I had this crazy idea trying to prove the Pythagorean trigonometric identity;$$\sin^2x+\cos^2x=1$$by squaring the infinite Taylor series of $\sin x$ and $\cos x$. But it came out quite beautiful, involving also a combinatorics identitie. The…
76david76
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Is Fourier series an "inverse" of Taylor series?

I've understood Taylor series as being the representation of a "transcendental" function, using power functions with coefficents represented by appropriate derivatives. (Or maybe it is the MacLauren series, where $\cos x=…
Tom Au
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