Taylor Definition and 878 Threads

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. N

    Why Do Theorists Use Series Expansion in Lagrangian Models?

    Hi, I have a following question... Can it be that there is given some Lagrangian and instead of considering whole Lagrangian one makes its series expansion and considers only some orders of expansion? Can you bring some examples or why and when does this happen... ? Thank you
  2. S

    Does the Taylor series expansion for e^x converge quickly?

    Hello all, My question is in regards to the Taylor series expansion of f(x)=e^x=1+x+x^2/(2!)+x^3/(3!)... I calculated the value of e^(-2) using the first 4 terms, 6 terms, and then the first 8 terms. I then calculated the relative error to compare it to the true value, depcited by my...
  3. K

    What is the Taylor Series Approximation for f(x)=(x0.5-1)/0.5 and f(x)=(x-1)2?

    Homework Statement Hi! I have a couple of problems on Taylor Series Approximation. For the following equations, write out the second-order Taylor‐series approximation. Let x*=1 and, for x=2, calculate the true value of the function and the approximate value given by the Taylor series...
  4. S

    Taylor Series/Newton Raphson Method question -Link Fixed

    http://uploadpic.org/storage/2011/dGTvcFGgvl4VYoVB40HpLSMxH.jpeg Can somebody guide me through this? I know how to apply Newton Raphson Method, but the x^* symbol and "argmin" function are kinda new to me. I am re referring to part (c). Thanks.
  5. L

    Taylor's Theorem for Sin(a+x) and Proving Convergence | Homework Solution

    Homework Statement Taylor's theorem can be stated f(a+x)=f(a)+xf'(a)+(1/2!)(x^2)f''(a)+...+(1/n!)(x^n)Rn where Rn=fn(a+y), 0≤y≤x Use this form of Taylor's theorem to find an expansion of sin(a+x) in powers of x, and show that in this case, mod(\frac{x^n Rn}{n!})\rightarrow0 as...
  6. S

    Taylor Series/Newton Raphson Method question

    Can somebody guide me through this? I know how to apply Newton Raphson Method, but the x^* symbol and "argmin" function are kinda new to me. I am re referring to part (c). Thanks.
  7. H

    Taylor Series of Log(z) around z=-1+i

    Homework Statement Find the taylor series of Log(z) around z=-1+i.Homework Equations The Attempt at a Solution So I have for the first few terms as \frac{1}{2}*log(2)+\frac{3\pi i}{4}+\frac{z+1-i}{-1+i}-\frac{2(z+1-i)^{2}}{(-1+i)^{2}}+\frac{3(z+1-i)^{3}}{(-1+i)^{3}}- But the correct...
  8. G

    Taylor Polynomials: Exploring Different Derivations

    http://bildr.no/view/1030479 The link above, it is my own and it is a bit disorderly, I think should explain taylor polynomials. In one assignent one had an assignment to derive taylor polynomials for cost^2 If one use the derivation rules with chain one get 2t for first derivative and...
  9. P

    What is the correct Taylor expansion for sin x around -pi/4 to the fourth term?

    I am asked to solve the taylor expansion of sin x around the point -pi/4 to the fourth term. I got sin(-pi/4)+cos(-pi/4)(x+pi/4)-.5sin(-pi/4)(x+pi/4)^2-1/6(cos(-pi/4)(x+pi/4)^3 but I am getting it wrong and can't see my mistake.
  10. S

    3rd order, multivariable taylor series

    Homework Statement Hello all, I have been working on a 3rd order taylor series, but the formula I have does not seem to get me the right answer. The formula I was given is for a taylor polynomial about point (a,b) is: P_3=f(a,b) +\left( f_{1}(a,b)x+f_{2}(a,b)y\right)...
  11. D

    Calculating Taylor Series for e^(x^2) around x=0

    Homework Statement Find the Taylor series of e^(x^2) about x=0 Homework Equations Taylor Series = f(a) +f'(a)(x-a) + (f''(a)(x-a)^2)/2 ... The Attempt at a Solution So, the first term is pretty obvious. It's e^0^2, which is zero. The second term is what got me...
  12. G

    Laurent and Taylor series in the unit disc

    Homework Statement Let f(z) be a function that is analytic for all |z|≤1, with the exception of z_0, which lies on the circle |z|=1. f(z) has a first order pole at z_0. Letting Ʃ a_n z^n be the Maclaurin expansion of the function, prove that z_0 = lim_(n→∞) a_n/a_(n+1) Homework Equations...
  13. T

    Taylor Series Expansion to Compute Derivatives

    Homework Statement Find the Taylor series expansion of f(x) = (x-1)/(1+(x-1)^2) about x=1 and use this to compute f(9)(1) and f(10)(1) Homework Equations The sum from n=0 to infinity of f(k)(c)/(k!) (x-c)k The Attempt at a Solution I'm not sure how to approach this...
  14. L

    Solve Taylor Series & Groups Homework: Show T(a) = exp(iap_x)

    Homework Statement A translation operator T(a) coverts ψ(x) to ψ(x+a), T(a)ψ(x) = ψ(x+a) In terms of the (quantum mechanical) linear momentum operator p_x = -id/dx, show that T(a) = exp(iap_x), that is, p_x is the generator of translations. Hint. Expand ψ(x+a) as a Taylor series...
  15. N

    Taylor serie of a function 1/(1+Z^2)

    Hello folks, I have this function, un complex numbers \frac{1}{(1+z^2)} I know that the Taylor serie of that function is \frac{1}{(1+z^2)} = \sum (-1)^k.z^(2.k)
  16. M

    Proof involving Taylor Polynomials / Lagrange Error Bound

    Homework Statement I'm given that the function f(x) is n times differentiable over an interval I and that there exists a polynomial Q(x) of degree less than or equal to n s.t. \left|f(x) - Q(x)\right| \leq K\left|x - a\right|^{n+1} for a constant K and for a \in I I am to show that Q(x)...
  17. G

    Proof Taylor Formula: Find without Polynomial Rules

    Does anyone know a proof of taylor formula (actually I am looking for proof for maclaurin series but guess it is the same) without using derivation rules for polynomials?
  18. D

    Hyperbolic sine in Taylor Series

    I am reading through a worked example of the Taylor series expansion of Sinh(z) about z=j*Pi The example states: sinh(j*Pi)=cos(Pi)*Sinh(0) +jcosh(x)sin(y) I am unsure of this relation. I understand why the x terms are zero but don't know the relation to expand sinh. Can anyone shed...
  19. Shackleford

    Analysis: Taylor Polynomial Approximation

    For #4, I'm mostly confident I did it correctly. In determining the error, we're supposed to find the maximum absolute value on an interval I. I set I = (0,2pi). Is that right? http://i111.photobucket.com/albums/n149/camarolt4z28/4-1.png For #5...
  20. H

    A question about Taylor series expansion

    Homework Statement Find the Taylor series expansion for f(x)=x*e^(-x^2) about x = -1 Homework Equations The Attempt at a Solution I have tried replacing x with (x-1) and f(x-1) = (x-1)*e^(-(x-1)^2). Consider the power series for e^(-(x-1)^2) about x = 0, f(x-1) =...
  21. H

    A question about Taylor series expansions

    Find the Taylor series expansions for f(x)=x*e^(-x^2) about x = -1 -(1/E) - (x + 1)/E + (x + 1)^2/E + (5 (x + 1)^3)/(3 E) + (x + 1)^4/( 6 E) - (23 (x + 1)^5)/(30 E) - (29 (x + 1)^6)/(90 E) + ( 103 (x + 1)^7)/(630 E)... This is the answer from Mathematica but i don't know how it goes. Can...
  22. S

    How to get (Taylor) series formula for arcosh?

    I'm looking at the series published @ Wikipedia: http://en.wikipedia.org/wiki/Inverse_hyperbolic_function There is a series for arsinh, which I was able to derive with no problem - basically take the derivative of arsinh, which is a radical, then apply the general binomial expansion, which...
  23. L

    Summing a series- Taylor series/ complex no.s?

    Homework Statement Find the sums of the following series: S1=1+(x^3)/(3!)+(x^6)/(6!)+... S2=x+(x^4)/(4!)+(x^7)/(7!)+... S3=(x^2)/(2!)+(x^5)/(5!)+(x^8)/(8!)+... Homework Equations Perhaps Taylor series? The Attempt at a Solution I spotted that adding S1+S2+S3=e^x, but I don't...
  24. J

    Taylor Polynomial with Remainder Question

    Homework Statement What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places? Homework Equations R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial) The Attempt at a Solution I have attached my attempt. I am stuck on the...
  25. A

    Taylor Series Expansion for f(z) = −1/z^2 about z = i + 1

    Homework Statement Find the Taylor series expansions for f(z) = −1/z^2 about z = i + 1. Homework Equations The Attempt at a Solution I'm just not sure what format I'm supposed to leave it in. Is it meant too look like this: f(z)=f(i+1)+f'(i+1)(x-i-1)... or this Ʃ\frac{1}{n!}f^{(n)}(1+i) *...
  26. N

    Taylor series and second derivative test: the degenerate case.

    Hello! I am wondering if someone could let me know if my understanding is right or wrong. The Taylor series gives the function in the form of a sum of an infinite series. From this an approximation of the change in the function can be derived: f_{a} and f_{a,a} are the first and second...
  27. A

    What is the trick to simplifying the Taylor series of 1/(1 + x^2)?

    The equation starts at B and this is my attempt. As you can see it soon complicates and doesn't look like what t should since I already know what the Taylor series of his function should look like. Is there some clever trick to it that I am missing? PS the series is centred around x = 0...
  28. T

    Taylor series at a point for which the function isn't defined (perturbation)

    Homework Statement This problem arises from the following ODE: \epsilon y'' + y' + y = 0, y(0) = \alpha, y(1) = \beta where 0 < x < 1, 0 < \epsilon \ll 1 Find the exact solution and expand it in a Taylor series for small \epsilon Homework Equations I guess knowing the Taylor...
  29. O

    Taylor Series: Show Terms Decay as 1/n^2

    Show that, with an appropriate choice of constant c, the taylor series of (1+cx)ln(1+x) has terms which decay as 1/n^2 I know that ln(1+x) decays as 1/n, but I don't know how to show the above. Please help. Thanks in advance
  30. S

    Proving Quadratic Convergence via Taylor Expansion

    Homework Statement The following is a modification of Newton's method: xn+1 = xn - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn) Homework Equations We are supposed to use the following method: let En = xn + p where p = root → xn = p + En Moreover, f(xn) = f(p + En) = f(p) +...
  31. P

    Confused about Taylor and Maclaurin Series

    Currently, I'm doing some self studying on series, and I'm a bit confused regarding c (the value that the series is expanded about). For example, does the Maclaurin series expansion of Sin(x) and the Taylor series of Sin(x) about c = 1 both converge to Sin(x)? If so, what does the value...
  32. D

    How can the Taylor series help prove the limit of cosine?

    I have to prove that \cos(x) = 1 - \frac{x^2}{2} + O(x^4) (x \to 0) My ugly attempt: \lim_{x \to 0} \frac{\cos(x) - 1 + \frac{x^2}{2}}{x^4} \lim_{x \to 0} \frac{\cos(x) - 1}{x^4} + \frac{1}{2x^2} \lim_{x \to 0} \frac{\sin(x)}{4x^3} + \frac{1}{2x^2} \lim_{x \to 0}...
  33. E

    Taylor Polynomial Approximations.

    Hello, I'm new here, nice to meet you guys i was in class today and just didn't understand the taylor polynomial approximation, the professor started out approximating a function by polynomials of degree N, he first showed us how a linear polynomial was a crude approximation of the function...
  34. J

    Using Taylor Polynomial for Laplace Transforms

    Ive attached the problem and my work in the pic. Questions: Am I even applying the taylor polynomial the correct way? (I never learned taylor series, but I was supposed to be taught in the pre-requisite class) Am I suppose to plug in c=4? I am not so sure about how the U4(t) works...
  35. Q

    Taylor expansion centering question

    what does it mean to say taylor expansion of ex centered at 0? does it mean that the sum of the expansion will give me the value that the function ex will take when x = 0 ? so its e0 = 1? also, how do we know what value to center on? because as i encounter taylor series in my...
  36. W

    Taylor expansion, of gradient of a function, in multiple dimensions

    Hello all, I understand that the taylor expansion for a multidimensional function can be written as f(\overline{X} + \overline{P}) = f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P}) where t is on (0,1). Although I haven't seen that form before, it makes sense...
  37. J

    Finding Taylor Series - different Method

    Homework Statement Hello, I'm in the middle of solving for the Taylor series of the function: f(x)=sin(2x)ln(1-x) up to n = 4. The Attempt at a Solution So far, I've been strictly taking its derivatives until I reach the fourth. It's becoming a very long process considering it's...
  38. P

    Taylor Polynomial Approximations (Apostol Section 7.8 #7)

    Homework Statement Prove that 0.493948<\int_0^{1/2} \frac{1}{1+x^4} dx<0.493958Homework Equations This chapter is about Taylor Polynomials, and specifically this section deals with Taylor's formula with remainder: f(x)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k + E_n(x) The general formula for...
  39. N

    Complex analysis, taylor series, radius of convergence

    Homework Statement For f(z) = 1/(1+z^2) a) find the taylor series centred at the origin and the radius of convergence. b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius. Homework...
  40. B

    Taylor's Upper Bound: f(x) 2x Diff. Function (0,∞)

    upper bound of taylor! f(x) is two times diff. function on (0, \infty) . \lim\limits_{x\rightarrow \infty}f(x) = 0 satisfy. M=\sup\limits_{x>0}\vert f^{\prime \prime} (x) \vert satisfy . for each integer L , g(L) = \sup\limits_{x\geq L} \vert f(x) \vert, and h(L) = \sup\limits_{x\geq L} \vert...
  41. R

    Taylor series and the forward finite difference method

    Given a partial differential equation, how would one go about implementing the forward finite difference method to the Taylor series?
  42. C

    Taylor Series : How to determine coefficient

    Homework Statement https://skydrive.live.com/?cid=6b041751c72e14ad#!/?cid=6b041751c72e14ad&sc=photos&uc=3&id=6B041751C72E14AD%21149!cid=6B041751C72E14AD&id=6B041751C72E14AD%21154&sc=photos The Attempt at a Solution...
  43. R

    Using Taylor Series for Initial Value Problems

    Homework Statement I posted this already but decided to revive this thread since I re-worked the problem. Consider dy/dx=x+y, a function of both x and y subject to initial condition, y(x0)=y0. Use Taylor series to determine y(x0+\Deltax) to 4th order accuracy. Initial condition: x0=0...
  44. M

    Taylor expansion in radial coordinates

    Homework Statement This shouldn't be so hard to do I guess, but I just cannot figure it out. The problem statement: Prove that the special form of the discrete Laplacian operator in radial coordinates acting on a grid function u_{l,m} at the central grid point l=0, m=0, given by...
  45. T

    Expansion of Taylor series for statistical functionals

    Hi By some googling it seems like there exist some kind of expansion of the Taylor series for statistical functionals. I can however, not sort out how it is working and what the derivative-equivalent of the functional actually is. My situation is that I have a functional, say \theta which...
  46. G

    Calculus II - Taylor Series Question

    Homework Statement Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) Homework Equations The Attempt at a Solution Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) f(x) =...
  47. G

    How Accurate is the Taylor Polynomial for Approximating Sin Functions?

    New Question (Changed Old one) - Taylor Polynomial - Upper Bound for Absolute Error Homework Statement (a) Find the 3-rd degree Taylor polynomial of sin(pix) centered at x=1. (b) Use (a) to approximate sin(1.1*pi) (c) Use the remainder term to find an upper bound for the absolute error in...
  48. G

    Calculus II - Taylor Series - Error Bounds

    Homework Statement Hi, I'm really struggling with trying to come up with the error bound when doing taylor series problems Use the reaminder term to estimate the absolute error in approximating the following quantitites with the nth-order Taylor Polynomial cnetered at 0. Estimates are...
  49. C

    How Do You Calculate Taylor Polynomials for f(x,y) = ln(3y-8x) at Point (1,1)?

    Homework Statement f(x,y) = ln(3y-8x) Derive the first and second order Taylor polynomial approx, L(x,t) and Q(x,t), for T(x,T) about the point (1,1) Homework Equations -None- The Attempt at a Solution I do not understand what the question wants, nor do i want a solution. I...
  50. V

    Examining the Taylor Series - Confused?

    hello, I'm examinating the theorem of power series, specially taylor series I know a function f(x) can be written as a series of polynomials. but using the taylor series it says that the convergence of that function is about a point a by using the Maclaurinseries a = 0 , so examinating...
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