What is Taylor: Definition and 873 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. G

    Help me with Taylor's theorem please

    I am trying to grasp how the last equation is derived. I understand everything, but the only thing problematic is why in the end, it's ##+O(\epsilon)## and not ##-O(\epsilon)##. It will be easier to directly attach the image, so please, see image attached.
  2. Reuben_Leib

    I Why does ##u## need to be small to represent the Taylor expansion

    Necessary condition for a curve to provide a weak extremum. Let ##x(t)## be the extremum curve. Let ##x=x(t,u) = x(t) + u\eta(t)## be the curve with variation in the neighbourhood of ##(\varepsilon,\varepsilon')##. Let $$I(u) = \int^b_aL(t,x(t,u),\dot{x}(t,u))dt = \int^b_aL(t,x(t) +...
  3. L

    I Taylor Expansion Question about this Series

    Can you please explain this series f(x+\alpha)=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^nf}{dx^n} I am confused. Around which point is this Taylor series?
  4. rdemyan

    A How Does Anisotropy Affect the Calculation of Taylor Microscale in Turbulence?

    The Taylor microscale in isotropic turbulence is given by: $$\lambda = \sqrt{ 15 \frac{\nu \ v'^2}{\epsilon} }$$ where v' is the root mean square of the velocity fluctuations. In general, for velocity fluctuations in three dimensions: $$v' =...
  5. A

    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)##?
  6. esrever10

    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?
  7. bigmike94

    Prerequisites for John Taylor classical mechanics

    Inside the textbook, the prerequisites state first year mechanics and some differential equations, although it continues to say the differential equations can be learned as you’re working your way through the book, as differential equations were basically “invented” to be used for applied...
  8. 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...
  9. bigmike94

    I Topics covered in John R Taylor Classical mechanics

    I can’t find the chapter list online, does anyone know what topics are covered in John Taylor’s classical mechanics? Would it be similar to what’s covered in Newtonian mechanics, but obviously more advanced. Cheers in advance 👍
  10. mcastillo356

    B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?

    Hi, PF For example, ##\sin{x}=O(x)## as ##x\rightarrow{0}## because ##|\sin{x}|\leq{|x|}## near 0. This fits textbook definition; easy, I think. But, Taylor's Theorem says that if ##f^{(n+1)}(t)## exists on an interval containing ##a## and ##x##, and if ##P_{n}## is the ##n##th-order Taylor...
  11. S

    Taylor Polynomials question

    f(x) = 4 + 5x - 6x^2 + 11x^3 - 19x^4 + x^5 question a almost seems too easy as I end up 'removing' the x^4 and x^5 terms a. T_{2} (x) = 4 + 5x - 6x^2 b. = R_{2} (x) = 11x^3 - 19x^4 + x^5 c. i don't understand what i need to do here. To find the maximum value of a function, we...
  12. A

    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...
  13. A

    Problem with a taylor serie expansion

    Greetings https://www.physicsforums.com/attachments/295843 I really don´t agree with the solution https://www.physicsforums.com/attachments/295846 as I calculated fxy I got fxy=xyexy f(0,1)=0 so x(y-1) should not appear in the solution am I wrong? thank you!
  14. L

    Multiplication of Taylor and Laurent series

    First series \frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}-\frac{1}{2p^4}+\frac{1}{3p^6}-\frac{1}{4p^8}+...) whereas second one is...
  15. Z

    How to choose the correct function to use for a Taylor expansion?

    Consider two different Taylor expansions. First, let ##f_1(s)=(1+s)^{1/2}## $$f_1'(s)=-\frac{1}{2(1+s^{3/2})}$$ Near ##s=0##, we have the first order Taylor expansion $$f_1(s) \approx 1 - \frac{s}{2}$$ Now consider a different choice for ##f(s)## $$f_2(s)=(1+s^2)^{1/2}$$...
  16. Mayhem

    I How do I use induction more rigorously when making Taylor expansions?

    When I do Taylor expansions, I take the first 3 or 4 derivatives of a function and try to induce a pattern, and then evaluate it at some value a (often 0) to find the coefficients in the polynomial expansion. This is how my textbook does it, and how several other online sources do it as well...
  17. B

    Understanding Taylor Expansion near a Point

    I'm just trying to understand how this works, because what I've been looking at online seems to indicate that I evaluate at ##\delta =0## for some reason, but that would make the given equation for the Taylor series wrong since every derivative term is multiplied by some power of ##\delta##...
  18. S

    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|...
  19. L

    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)?
  20. B

    4th order Taylor approximation

    So I just followed Taylor's formula and got the four derivatives at p = 0 ##f^{(0)}(p) = (1 + \frac {p^2} {m^2c^2})^{\frac 1 2} ## ##f^{(0)}(0) = 1 ## ## f^{(1)}(p) = \frac {p} {m^2c^2}(1 + \frac {p^2} {m^2c^2})^{\frac {-1} 2} ## ## f^{(1)}(0) = 0 ## ## f^{(2)}(p) = \frac {1} {m^2c^2}(1 +...
  21. C

    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...
  22. A

    Problem with series convergence — Taylor expansion of exponential

    Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!
  23. A

    I Taylor expansion of an unknown function

    Hello, I have a question regarding the Taylor expansion of an unknown function and I would be tanksful to have your comments on that. Suppose we want to find an analytical estimate for an unknown function. The available information for this function is; its exact value at x0 (f0) and first...
  24. Hiero

    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...
  25. T

    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...
  26. T

    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...
  27. PainterGuy

    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...
  28. W

    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...
  29. S

    Taylor expansion of an Ising-like Hamiltonian

    For the case when ##B=0## I get: $$Z = \sum_{n_i = 0,1} e^{-\beta H(\{n_i\})} = \sum_{n_i = 0,1} e^{-\beta A \sum_i^N n_i} =\prod_i^N \sum_{n_i = 0,1} e^{-\beta A n_i} = [1+e^{-\beta A}]^N$$ For non-zero ##B## to first order the best I can get is: $$Z = \sum_{n_i = 0,1}...
  30. C

    I Question about Taylor Expansions

    I was working out a problem requiring a taylor expansion of ## \sqrt {1+x^2} ## (about ##x=0##). I needed to go out to the 5th term in the expansion, which, while not difficult, was long and annoying as the ##x^2## necessitated chain rules and product rules when taking the derivatives and the...
  31. L

    Using a Taylor expansion to prove equality

    Homework Statement: Use Taylor expansion to show that for ##u \in C^4([0,1]) ## $$ max |\partial^+\partial^-u(x) - u''(x)| = \mathcal{O}(h^2)$$ For ##x \in [0,1]## and where the second order derivative ##u''## can be approximated by the central difference operator defined by...
  32. S

    Python Finding the coefficients of a Taylor polynomial

    To find the coefficients of the Taylor polynomial of degree two of the function ##z(x,y)## around the point ##(0,0)##, what would be a handy way of doing that in python? How would one find the derivatives of ##z(x,y)##?
  33. Jason Bennett

    (Physicist version of) Taylor expansions

    3) Taylor expansion question in the context of Lie algebra elements: Consider some n-dimensional Lie group whose elements depend on a set of parameters \alpha =(\alpha_1 ... \alpha_n) such that g(0) = e with e as the identity, and that had a d-dimensional representation D(\alpha)=D(g( \alpha)...
  34. JorgeM

    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...
  35. Adgorn

    Limit of the remainder of Taylor polynomial of composite functions

    Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...
  36. E

    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...
  37. CricK0es

    Expanding a function for large E using the Taylor Expansion technique

    I have been playing around with Taylor expansion to see if I can get anything out but nothing is jumping out at me. So any hints, suggestions and preferably explanations would be greatly appreciated as I’ve spent so so long messing around with it and I need to move on... But as always, thank you
  38. m4r35n357

    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...
  39. silverfury

    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
  40. PainterGuy

    I How to Derive the 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...
  41. Physics lover

    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.
  42. S

    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} =...
  43. S

    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...
  44. m4r35n357

    Damped & Driven Pendulums (in _pure_ Python)

    This is another application of using Taylor recurrences (open access) to solve ODEs to arbitrarily high order (e.g. 10th order in the example invocation). It illustrates use of trigonometric recurrences, rather than the product recurrences in my earlier Lorenz ODE posts. Enjoy! #!/usr/bin/env...
  45. mertcan

    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...
  46. A

    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...
  47. W

    I 1D scattering: Taylor expansion

    Hi all, I'm having a problem understanding a step in an arxiv paper (https://arxiv.org/pdf/0808.3566.pdf) and would like a bit of help. In equation (29) the authors have $$R = \frac{\sigma}{\sqrt{\pi}} \int dk \ e^{-(k - k_0)^2 \sigma^2} \ \Big( \frac{ k - \kappa}{ k+ \kappa} \Big)^2$$ where...
  48. S

    How to Evaluate the 8th Derivative of a Taylor Series at x=4

    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...
  49. SebastianRM

    I What is the tangential component? Taylor p.347

    I do not understand why the tan component for a gravity affected by the centrifugal force: g = Ω^2 * R * sinθ * cosθ So I tried to draw this: using a "big" X-shaped axis where the / component goes along the main gravity direction while \ points normal to / this direction. Then the centrifugal...
  50. SpaceIsCool

    Transverse velocity and real/imaginary parts?

    Homework Statement The transverse velocity of the particle in Sections 2.5 and 2.7 is contained in (2.77), since By taking the real and imaginary parts, find expressions for v_x and v_y separately. Based on these expressions describe the time dependence of the transverse velocity. Homework...
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