# What is Taylor series: Definition and 491 Discussions

In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.
If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

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1. ### Questions about what a question is asking

In my introductory modern physics class, I was asked to compute the Taylor Series for exp(-ax^2) about the value x = 0 to second order in x. I am unfamiliar with the what "exp" before the function means, despite having approximated functions with Taylor Series before. I think there was some gap...
2. ### B Taylor Series of e^(f(x,y))

I know that for 1 variable, one can write ##e^{f(x)} = \sum_{n = 0}^{\infty}\frac{(f(x))^n}{n!}##. In the case of 2-variables ##f(x,y)##, I assume we cannot write ##e^{f(x,y)} = \sum_{n = 0}^{\infty}\frac{(f(x,y))^n}{n!}## right (because of how the Taylor series is defined for multiple...
3. ### I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?

Here is a proof of mean value theorem: Consider a line passing through the points (a, f(a)) and (b, f(b)). The equation of the line is y-f(a) = {(f(b)-f(a))/(b-a)} (x-a) or y = f(a)+ {(f(b)-f(a))/(b-a)} (x-a) Let h be a function that defines the difference between any function f and the...
4. ### Doubt about the derivative of a Taylor series

My doubt arises over the definition of L'(v^2). If we are using ##x= v'^2##, shouldn't the derivative be made with respect to that very term? In essence, shouldn't it be: L'(v^2) = \frac{\partial L(v^2)}{\partial (v'^2)}? In the article I read, L'(v^2) = \frac{\partial L(v^2)}{\partial (v^2)} is...
5. ### Show that the Taylor series for this Lagrangian is the following...

We have ##L(v^2 + 2v\epsilon + \epsilon^2)##. Then, the book proceeds to mention that we need to expand this in powers of ##\epsilon## and then neglect the terms above first order, we obtain: ##L(v^2) + \frac{\partial L}{\partial v^2}2v\epsilon## (This is what I don't get). We know taylor is...
6. ### I Intuiting Lieb-Robinson Bound

Hi all, I was wondering if there was a way to intuit the Lieb-Robinson bound from simply looking at the taylor series for an operator ##A(t) = e^{-iHt}Ae^{iHt}## where ##H## is a k-local Hamiltonian and ##A(t)## initially starts off as a single-site operator. The generic idea is that at each...
7. ### I Can we solve a non-autonomous diffeq via Taylor series?

I've occasionally seen examples where autonomous ODE are solved via a power series. I'm wondering: can you also find a Taylor series solution for a non-autonomous case, like ##y'(t) = f(t)y(t)##?
8. ### Taylor Series Expansion of f(x) at 0

First I got ##f(0)=0##, Then I got ##f'(x)(0)=\frac{\cos x(2+\cosh x)-\sin x\sinh x}{(2+\cosh x)^2}=1/3## But when I tried to got ##f''(x)## and ##f'''(x)##, I felt that's terrible, If there's some easy way to get the anwser?
9. F

### Link between Z-transform and Taylor series expansion

Hello, I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused. Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...
10. ### Finding the Taylor series of a function

Greetings! Here is the solution that I understand very well I reach a point I think the Professor has mad a mistake , which I need to confirm after putting x-1=t we found: But in this line I think there is error of factorization because we still need and (-1)^(n+1) over 3^n Thank you...
11. ### Bounds of the remainder of a Taylor series

I have found the Taylor series up to 4th derivative: $$f(x)=\frac{1}{2}-\frac{1}{4}(x-1)+\frac{1}{8}(x-1)^2-\frac{1}{16}(x-1)^3+\frac{1}{32}(x-1)^4$$ Using Taylor Inequality: ##a=1, d=2## and ##f^{4} (x)=\frac{24}{(1+x)^5}## I need to find M that satisfies ##|f^4 (x)| \leq M## From ##|x-1|...
12. ### A Limits of Taylor Series: Is $\sin x=x+O(x^2)$ Correct?

We sometimes write that \sin x=x+O(x^3) that is correct if \lim_{x \to 0}\frac{\sin x-x}{x^3} is bounded. However is it fine that to write \sin x=x+O(x^2)?
13. ### Can you use Taylor Series with mathematical objects other than points?

I was recently studying the pressure gradient force, and I found it interesting (though this may be incorrect) that you can use a Taylor expansion to pretend that the value of the internal pressure of the fluid does not matter at all, because the internal pressure forces that are a part of the...
14. ### I Looking for references on this form of a Taylor series

I was trying to find this form of the Taylor series online: $$\vec f(\vec x+\vec a) = \sum_{n=0}^{\infty}\frac{1}{n!}(\vec a \cdot \nabla)^n\vec f(\vec x)$$ But I can’t find it anywhere. Can someone confirm it’s validity and/or provide any links which mention it? It seems quite powerful to be...
15. ### Evaluate the Taylor series and find the error at a given point

I have the following function $$f^{(0)}\left(x\right)=f\left(x\right)=e^{x}$$ And want to approximate it using Taylor at the point ##\frac{1}{\sqrt e} ## I also want to decide (without calculator)whether the error in the approximation is smaller than ##\frac{1}{25} ## The Taylor polynomial is...
16. ### MHB How to estimate simplex gradient using Taylor series?

I read Iterative methods for optimization by C. Kelley (PDF) and I'm struggling to understand proof of Notes on notation: S is a simplex with vertices x_1 to x_{N+1} (order matters), some edges v_j = x_j - x_1 that make matrix V = (v_1, \dots, v_n) and \sigma_+(S) = \max_j \lVert...
17. ### I Questions about this video on Taylor series

Hi, I was watching a video on the origin of Taylor Series shown at the bottom. Question 1: The following screenshot was taken at 2:06. The following is said between 01:56 - 02:05: Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one...
18. ### I Taylor series and variable substitutions

I'm currently typing up some notes on topics since I have free time right now, and this question popped into my head. Given a problem as follows: Find the first five terms of the Taylor series about some ##x_0## and describe the largest interval containing ##x_0## in which they are analytic...
19. ### I 2nd order Taylor Series for a function in 3 or more variables?

I have taken a look but most books and Online stuff just menctions the First order Taylor for 3 variables or the 2nd order Taylor series for just 2 variables. Could you please tell me which is the general expression for 2nd order Taylor series in 3 or more variables? Because I have not found...
20. ### Find the Maximum of a Multi-variable Taylor Series

Firstly, the matrix notation of the series is, \begin{align*} f\left(x, y, z\right) &= f\left(a, b, c\right) + \left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial f\left(a, b, c\right)}{\partial \mathbf{x}} + \frac{1}{2}\left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial^2 f\left(a, b...
21. ### Python The Wimol-Banlue attractor via the Taylor Series Method

This attractor is unusual because it uses both the tanh() and abs() functions. A picture can be found here (penultimate image). Here is some dependency-free Python (abridged from the GitHub code, but not flattened!) to generate the data to arbitrary order: #!/usr/bin/env python3 from sys...
22. ### Can someone help me with this Taylor series expansion?

I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help
23. ### I Taylor series for log(x)

Hi, I was trying to solve the following problem myself but couldn't figure out how the given Taylor series for log(x) is found. Taylor series for a function f(x) is given as follows. Question 1: I was trying to find the derivative of log(x). My calculator gives it as...
24. ### A limit problem without the use of a Taylor series expansion

I tried substituting x=cos2theta but it was of no use.I thought many ways but i could not make 0/0 form.So please help.
25. ### B Is there a condition for applying standard limits?

I came across this basic limits question Ltx->0[(ln(1+X)-sin(X)+X2/2]/[Xtan(X)Sin(X)] The part before '/'(the one separated by ][ is numerator and the one after that is denominator The problem is if I substitute standard limits : (Ltx->0tan(X)/x=1 Ltx->0sin(X)/X=1 Ltx->0ln(1+X)/X=1) The...
26. ### Find the Taylor series of a function

Because the Taylor series centered at 0, it is same as Maclaurin series. My attempts: 1st attempt \begin{align} \frac{1}{1-x} = \sum_{n=0}^\infty x^n\\ \\ \frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n\\ \\ \frac{1}{x^2} = \sum_{n=0}^\infty (1-x^2)^n\\ \\ \frac{1}{(2-x)^2} =...
27. ### Lagrange error bound inequality for Taylor series of arctan(x)

The error ##e_{n}(y)## for ##\frac{1}{1-y}## is given by ##\frac{1}{(1-c)^{n+2}}y^{n+1}##. It follows that ##\frac{1}{1+y^2}=t_n(-y^2)+e_n(-y^2)## where ##t_n(y)## is the Taylor polynomial of ##\frac{1}{1-y}##. Taking the definite integral from 0 to ##x## on both sides yields that...
28. ### I How bad is my maths? (numerical ODE method)

I have written some ODE solvers, using a method which may not be well known to many. This is my attempt to explain my implementation of the method as simply as possible, but I would appreciate review and corrections. At various points the text mentions Taylor Series recurrences, which I only...
29. ### Taylor Series Convergence

Hi, as you know infinite sum of taylor series may not converge to its original function which means when we increase the degree of series then we may diverge more. Also you know taylor series is widely used for an approximation to vicinity of relevant point for any function. Let's think about a...
30. ### I Taylor Series for Potential in Crystals

Hi, I've been reading the passage attached below and from what I understand we are looking at a 1D chain of atoms and if anyone atom moves it changes the potential for surrounding atoms and cause a change in energy in the system so the total energy is dependent on all the positions of the atoms...
31. ### Taylor Series Evaluation

Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## Homework Equations The Taylor Series Equation The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
32. ### Approximating the force on a dipole Taylor series

Homework Statement Show that the magnitude of the net force exerted on one dipole by the other dipole is given approximately by:$$F_{net}≈\frac {6q^2s^2k} {r^4}$$ for ##r\gg s##, where r is the distance from one dipole to the other dipole, s is the distance across one dipole. (Both dipoles are...

38. ### I Taylor Series Expansion of Quadratic Derivatives: Goldstein Ch. 6, Pg. 240

Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below
39. ### I Which x_0 to use in a Taylor series expansion?

I already learn to use Taylor series as: f(x) = ∑ fn(x0) / n! (x-x0)n But i don´t see why the serie change when we use differents x0 points. Por example: f(x) = x2 to express Taylor series in x0 = 0 f(x) = f(0) + f(0) (x-0) + ... = 0 due to f(0) = (0)2 to x0=1 the series are...
40. ### How to represent a periodic function using Taylor series

Hi, Is this possible to represent a periodic function like a triangular wave or square wave using a Taylor series? A triangular wave could be represented as f(x)=|x|=x 0<x<π or f(x)=|x|=-x -π<x<0. I don't see any way of doing although I know that trigonometric series could be used instead...
41. ### A Transfinite Taylor series of exp(x) and of h(x)

Let h(h(x)) = exp(x), where h(⋅) is holomorphic in the whole ℂ plane. I want an extension of the domain of exp(⋅) and of h(⋅) so that we can find values of these functions for x = Aleph(0).
42. G

### Taylor Series for f(x) = (1+x)^m

Homework Statement i) What is the Taylor Series for f(x) = (1+x)^m about x=0 where m is a real number? ii) Why does this binomial series terminate when m is a non-negative integer? A iii) Can the result to (i) be used to find the first four non-zero terms of the series for (1+x)^(-1/2)...
43. ### Establish Taylor series using Taylor's Theorem in terms of h

Homework Statement Find the Taylor series for: ln[(x - h2) / (x + h2)] Homework Equations f(x+h) =∑nk=0 f(k)(x) * hk / k! + En + 1 where En + 1 = f(n + 1)(ξ) * hn + 1 / (n + 1)! The Attempt at a Solution ln[(x - h2) / (x + h2)] = ln(x-h2) - ln(x + h2) This is as far as I have been able to...

45. ### I Taylor series to evaluate fractional-ordered derivatives

Can the Taylor series be used to evaluate fractional-ordered derivative of any function? I got this from Wikipedia: $$\frac{d^a}{dx^a}x^k=\frac{\Gamma({k+1})}{\Gamma({k-a+1})}x^{k-a}$$ From this, we can compute fractional-ordered derivatives of a function of the form ##cx^k##, where ##c## and...
46. ### I Can Taylor series be used to get the roots of a polynomial?

I'm using this method: First, write the polynomial in this form: $$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x=c$$ Let the LHS of this expression be the function ##f(x)##. I'm going to write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be...
47. ### I Expression of ##sin(A*B)##

Hi, I've got this: $$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close). Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}## But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...
48. ### I What methods other than Taylor Series to solve this eq?

Hi If I have a problem of the form: A1ek1t + A2ek2t = C where A1,A2,k1,k2,C are real and known Or simplified: ex + AeBx = C I can turn it into an nth degree polynomial by Taylor Series expansion, but I'd like to know what other methods I can study Thanks, Archie
49. ### Is it possible to graph a function using its taylor series?

Homework Statement For example cosh(x) = 1+x2/2!+x4/4!+x6/6!+... Homework EquationsThe Attempt at a Solution So plugging in x=0 you get that coshx = 1 at the origin. The approximate graph for the coshx function up to the second order looks like a 1+x2/2! graph, but what about graphing coshx...
50. ### A Taylor series expansion of functional

I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field, L=½(∂φ)^2 - m^2 φ^2 in the equation, S[φ]=∫ d4x L[φ] ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2) Particularly, it is in the Taylor series...