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. vishal.ng

    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...
  2. Lujz_br

    Question 6.9 Taylor: Classical Mechanics

    Homework Statement Hello, I solved others but not 6.9: Find the equation of the path joining the origin O to point P(1,1) in the xy plane that makes the integral ∫(y'2 +yy' + y2) dx stationary. ∫ from O to P. y' = dy/dx Homework Equations I need use ∂f/∂y = d/dx (∂f/∂y') (euler-lagrange...
  3. doktorwho

    Find the limit using taylor series

    Homework Statement Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression: $$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$ Homework Equations 3. The Attempt at a Solution [/B] ##L=\lim_{x \rightarrow 0}...
  4. sa1988

    Taylor Expand Lagrangian to Second Order....

    Homework Statement NOTE - When I post the thread my embedded images aren't showing up on my web browser, but they do show up when I bring it up to edit, so I don't know if other users can see the pictures or not... If not, they're here: Problem outline: http://tinypic.com/r/34jeihj/9 Solution...
  5. C

    I Convergence of Taylor series in a point implies analyticity

    Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
  6. dykuma

    Convert Partial Fractions & Taylor Series: Solving Complex Equations

    Homework Statement and the solution (just to check my work) Homework Equations None specifically. There seems to be many ways to solve these problems, but the one used in class seemed to be partial fractions and Taylor series. The Attempt at a Solution The first step seems to be expanding...
  7. karush

    MHB 206.11.3.27 first three nonzero terms of the Taylor series

    $\textsf{a. Find the first three nonzero terms of the Taylor series $a=\frac{3\pi}{4}$}$ \begin{align} \displaystyle f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\sin{x} \\ f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\ f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=\frac{\sqrt{2}}{2} \\...
  8. karush

    MHB 206.11.3.39 Find the first four nonzero terms of the Taylor series

    $\tiny{206.11.3.39}$ $\textsf{a. Find the first four nonzero terms of the Taylor series $a=0$}$ \begin{align} \displaystyle f^0(x)&=(1+x)^{-2} &\therefore \ \ f^0(a)&= 1 \\ f^1(x)&=\frac{-2}{(x+1)^3} &\therefore \ \ f^1(a)&= -2 \\ f^2(x)&=\frac{6}{(x+1)^4} &\therefore \ \ f^2(a)&= 6 \\...
  9. karush

    MHB 242.13.3 Taylor remander formula

    $\tiny{242.13.3}$ $\textsf{1. Using the known series expansion of $\displaystyle e^x = \sum_{n=0}^{\infty}$, find the series representation of}\\$ $\textsf{a. $e^{-3x}$}\\$ $\textsf{b. $e^{x^3}$}$
  10. Kaura

    Taylor Series Error Integration

    Homework Statement Using Taylor series, Find a polynomial p(x) of minimal degree that will approximate F(x) throughout the given interval with an error of magnitude less than 10-4 F(x) = ∫0x sin(t^2)dt Homework Equations Rn = f(n+1)(z)|x-a|(n+1)/(n+1)![/B] The Attempt at a Solution I am...
  11. karush

    MHB 11. 1.33-T nth order Taylor polynomials - - centered at a=100, n=0

    $\tiny {11. 1.33-T} $ $\textsf{Find the nth order Taylor polynomials of the given function centered at a=100, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=\sqrt{x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n} \frac{f^{(k)}\left(a\right)}{k!}(x-a)^k$$ $\textsf{n=0}\\$...
  12. karush

    MHB Taylor Polynomials for $e^{-4x}$ at $x=0$

    $\tiny{206.11.1.16-T}$ $\textsf{Find the nth-order Taylor polynomials centered at 0, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=e^{-4x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$ $\textsf{n=0}\\$ \begin{align} f^0(x)&\approx...
  13. karush

    MHB Find the nth-order Taylor polynomials centered at 0, for n=0, 1, 2.

    $\tiny{206.11.1.15-T}$ $\textsf{Find the nth-order Taylor polynomials centered at 0, for $n=0, 1, 2.$}$ \\ $$\displaystyle f(x)=cos(3x)$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$ $\textsf{n=0}\\$...
  14. Kaura

    Taylor Series of cos(3x^2)

    Homework Statement Determine the Taylor series for the function below at x = 0 by computing P5(x) f(x) = cos(3x2) Homework Equations Maclaurin Series for degree 5 f(0) + f1(0)x + f2(0)x2/2! + f3(0)x3/3! + f4(0)x4/4! + f5(0)x5/5! The Attempt at a Solution I know how to do this but attempting...
  15. mastrofoffi

    Derivation of Taylor Series in R^n

    I was studying the derivation for taylor series in ℝ##^n## on my book and I have some trouble understanding a passage; it's the very beginning actually: ##f : A## ⊆ ℝ##^n## → ℝ ##f ## ∈ ##C^2(A)## ##x_0## ∈ ##A## "be ##g_{(t)} = f_{(x_0 + vt)}## where v is a generic versor, then we have...
  16. K

    Taylor mechanics ch3 problem7 -- Men jumping off of a railcar

    Homework Statement two man,each of equal mass m,are standing at one end of a stationary railroad flatcar with frictionless wheel and mass mcar.Find the car's speed if the two men run to the other end of the car and jump off simultaneously with the same speed u(relative to the car) Homework...
  17. K

    Taylor mechanics chap3,problem 4

    Homework Statement two man,each of equal mass m,are standing at one end of a stationary railroad flatcar with frictionless wheel and mass mcar.Find the car's speed if the two men run to the other end of the car and jump off simultaneously with the same speed u(relative to the car) Homework...
  18. K

    Performing a Taylor Series Expansion for Lorentz Factor

    Homework Statement Perform a Taylor Series expansion for γ in powers of β^2, keeping only the third terms (ie. powers up to β^4). We are assuming at β < 1. Homework Equations γ = (1-β^2)^(-1/2) The Attempt at a Solution I have no background in math so I do not know how to do Taylor expansion...
  19. Mr Davis 97

    First order term in the taylor expansion of ln(x) abut 1

    Homework Statement What's the first order term in the expansion ln(x) about x = 1? Homework Equations Taylor series formula The Attempt at a Solution The question is multiple choice, and the choices are x, 2x, or (1/2)x. However, when I calculate the first order term in the expansion of ln(x)...
  20. Amara

    Taylor expansion of the relativistic Doppler effect?

    [Note from mentor: this thread was originally posted in a non-homework forum, therefore it does not use the homework template.] I have been given an equation for the relativistic doppler effect but I'm struggling to see this as a function and then give a first order Taylor expansion. Any help...
  21. S

    I Linearizing vectors using Taylor Series

    I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector. Does that mean I will have to treat it as a Taylor expansion about two variables...
  22. T

    Taylor series representation for $$ \frac{x}{(1+4x)^2}$$

    Homework Statement Find a power series that represents $$ \frac{x}{(1+4x)^2}$$ Homework Equations $$ \sum c_n (x-a)^n $$ The Attempt at a Solution $$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$ since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} $$ x*\frac{d}{dx}\frac{1}{(1+4x)^2}...
  23. S

    I Is a function better approximated by a line in some regions?

    I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is...
  24. A

    I Taylor Series: What Is the Significance of the a?

    i watched a lot of videos and read a lot on how to choose it, but i what i can't find anywhere is, what's the physical significance of the a, if we were to draw the series, how will the choice of a affect it?
  25. S

    Find Taylor Series for 1/x Around x=3

    Homework Statement Find the Taylor Series for f(x)=1/x about a center of 3. Homework EquationsThe Attempt at a Solution f'(x)=-x^-2 f''(x)=2x^-3 f'''(x)=-6x^-4 f''''(x)=24x^-5 ... f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n I'm not sure where I went wrong...
  26. Eagertolearnphysics

    Classical Which is better Morin or Taylor on Classical Mechanics?

    I am a second year physics and I want to study CM in more depth than that of the general textbooks
  27. mertcan

    A Taylor expansion metric tensor

    hi, when I dug up something about metric tensors, I found a equation in my attached file. Could you provide me with how the derivation of this ensured? What is the logic of that expansion in terms of metric tensor? I really need your valuable responses. I really wonder it. Thanks in advance...
  28. tanvi nautiyal

    I Second order Taylor approximation

    Hello, Can someone explain this to me? In the above case ct=yt-gt I tried to solve it as a three variable taylor approximation but got a few extra terms that weren't included in the above. So I am a little confused now. I only need to understand how the first line was derived because I get...
  29. R

    Taylor Expansion: Computing x^2 + x^4/12

    Hello friends, I need to compute the taylor expansion of $$\frac{x^4 e^x}{(e^x-1)^2}, $$ for ##x<<1##, to find $$ x^2 + \frac{x^4}{12}.$$ Can someone explain this to me? Thanks!
  30. I

    Taylor Expansion For Scalar Field

    Homework Statement Page 35 of Jackson's Electrodynamics (3rd ed), it gives the following equation (basically trying to prove your standard 1/r potential is a solution to Poisson equation): \nabla^2 \Phi_a = \frac{ -1 }{ \epsilon_0 } \int \frac{ a^2 }{( r^2 + a^2)^{5/2} } \rho( \boldsymbol{x'}...
  31. binbagsss

    Chain rule / Taylor expansion / functional derivative

    Homework Statement To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0## where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)## Homework Equations I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...
  32. mertcan

    I Taylor expansion and parallel transport

    hi, first of all in this image there is a fact that we have parallel transported vector, and covariant derivative is zero along the "pr"path as you can see at the top of the image. I consider that p, and r is a point and in the GREEN box we try to make a taylor expansion of the contravariant...
  33. S

    Taylor polynom of f(x)=1/(√1-e3x)

    Hello, I can't find solution for Maclaurin (Taylor a=0) polynom of function: f(x)=1/(√1-e3x). Could you help me please? Thank you so much for help Andrea
  34. Junaid Aftab

    I Taylor and Wheeler's Spacetime Physics (1st Edition)

    Hi everyone, I've been trying to buy a copy of the first edition of the textbook "Spacetime Physics" by Taylor and Wheeler in my country, but I haven't been able to get my hands on a copy of it. Moreover, the e-books available online are poorly scanned with a bad font. I was able to download...
  35. A

    I Regarding Error Bound of Taylor Series

    Hi all, I am very confused about how one can find the upper bound for a Taylor series.. I know its general expression, which always tells me to find the (n+1)th derivative of a certain function and use the equation f(n+1)(c) (x-a)n+1/(n+1)! for c belongs to [a,x] However, there are...
  36. JulienB

    Taylor development I don't understand

    Homework Statement Hi everybody! In the middle of an exercise, our teacher suddenly wrote: sin(\frac{x}{y} sin y) = \frac{x}{y} sin y - \frac{1}{2} sin θ (\frac{x}{y} sin y)^2 I don't get where does that come from? The closest I've managed to reach is: sin(\frac{x}{y} sin y) =...
  37. G

    MHB How to find the upper bound of an error by Taylor polynomial approximation

    I'm struggling about finding a way to find the upper bound of the error of Taylor polynomial approximation. I will explain better using a solved example I found... > $f: ]-3;+\infty[ \rightarrow \mathbb{R} $ $f(x)=ln(x+3) +1 $ >Find the upper bound of the error approximating the function in...
  38. A

    MHB Simple to understand derivations similar to the Taylor Series

    That I don't even know in which forum to post this questions shows my gaping lack of mathematics knowledge. I've just learned the derivation of the Taylor series. I'm slapping myself on the head as it's so mind-bogglingly simple, but I never learned it. The Taylor series was just 'maths magic'...
  39. S

    How do I Taylor expand the gravitational field in terms of h/R <<1?

    Homework Statement Consider the position vector of a mass m at height h above the Earth's surface to be \underline{r}=(R+h)\underline{e}_z where R is the radius of the Earth. Make a Tylor expansion in h/R <<1 of the gravitational field...
  40. deagledoubleg

    Find Taylor Series from a function and its interval of convergence

    Let f(x) = (1+x)-4 Find the Taylor Series of f centered at x=1 and its interval of convergence. \sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!} is general Taylor series form My attempt I found the first 4 derivatives of f(x) and their values at fn(1). Yet from here I do not know how to find the...
  41. nfcfox

    The power series above is the Taylor series....

    Homework Statement http://imgur.com/1aOFPI7 PART 2 Homework Equations Taylor series form The Attempt at a Solution My thought process is that the answer is 3 because using the geometric series equation (1st term)/(1-R) then you can get the sum. In this case R would be x+2 where x is -2 so 0...
  42. P

    I Not understanding Hulse Taylor period shift calculation

    I have been studying Hulse Taylor PSR 1913+16 calculation of period shift which is regarded as indirect proof for gravitational waves, but I don't understand one thing. If you look on the graph of Cumulative period shift, around every 10 years the shift doubles...
  43. E

    Calculus Taylor Approximation Proof

    1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...
  44. Hepth

    Puiseux/Taylor Expansion of an Integrand pre-Integration

    My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero. I have a function ##f[x,r]## I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0## This is impossible algebraically for me, but may be possible if I can...
  45. I

    Checking Taylor Series Result of 6x^3-3x^2+4x+5

    Homework Statement Use zero- through third-order Taylor series expansion f(x) = 6x3 − 3x2 + 4x + 5 Using x0=1 and h =1. Once I found that the Taylor Series value is 49. I want to be able to check the value. On the board our teacher plugged in a value into the equation to show that the answer...
  46. I

    Proof that e is irrational using Taylor series

    Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...
  47. W

    Taylor polynomial/series, series, function series

    well, i have an calculus exam tomorrow and I'm 100% gona fail. I've neglected calculus so i could study for other subjects and left only 2 days to study taylor's polynomial aproximation, series and function series, the latter two are way more complicated than i expected. good thing is i can...
  48. S

    Taylor expansion of a scalar potential field

    Consider the potential ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##, where ##\phi## is a scalar field and the mass dimensions of the couplings are: ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##. Expanding the field ##\phi## about the point...
  49. B

    Taylor expansion with multi variables

    I was reading a book on differential equations when this(taylor expansion of multi variables) happened. Why does it not include derivatives of f in any form? The page of that book is in the file below.
  50. nomadreid

    Interval of convergence for Taylor series exp of 1/x^2

    Homework Statement The interval of convergence of the Taylor series expansion of 1/x^2, knowing that the interval of convergence of the Taylor series of 1/x centered at 1 is (0,2) Homework Equations If I is the interval of convergence of the expansion of f(x) , and one substitutes a finite...
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