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. P

    Comples analysis - Radius of convergence of a Taylor series question

    Homework Statement Find the radius of convergence of the Taylor series at z = 1 of the function: \frac{1}{e^{z}-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far. Multiply top and bottom by minus 1 to get: -1/(1-e^z) And then...
  2. S

    What functino has this Taylor Expansion?

    Something came up while I was trying to solve a problem. This is a Taylor Expansion of a function: x+(x^2)/2+(x^3)/3+(x^4)/4+(x^5)/5+(...) What's the function associated with it?
  3. B

    Approximating sin(1/10) using 3rd degree Taylor polynomial

    Homework Statement what is the 3rd degree taylor series of sin(1/10), and calculate the error of your answer. the wording of this question may be a little off, i just took a test and this was what i remembered about the question. The Attempt at a Solution i didnt think that this was...
  4. C

    Four proof that exercises about taylor polynomials

    Four "proof that" exercises about taylor polynomials Homework Statement Definition: A function f is called C^n if f n times derivable and if the n-time derivable f^(n) is continuos. If is from class C^n then its called f ε C^n Exercise 1) Be f ε C^n in the interval [a,x]. Be P a polynomial...
  5. MathematicalPhysicist

    Mathematica Letting mathematica compute taylor expansion of implicit function.

    I have the next function: z^3-2xz+y=0 and I want to find taylor expansion of z(x,y) at the point (1,1,1), obviously I need to define F(x,y,z) as above and use the implicit function theorem to calculate the derivatives of z(x,y), but I want mathematica to compute this to me. I tried the Series...
  6. N

    Taylor Polynomial of f(x) = x^3sin(x)

    Homework Statement Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = x^{3} sin(x) at a = 0. Homework Equations f^{n}(x) * (x-a)^{n} / (n!) The Attempt at a Solution I got the question wrong: my answer was 1/3! + 1/5! + 1/7! Here is the answer below. I...
  7. F

    MHB Exploring Taylor Series Expansions for Quadratic Equations

    Hi! I'm taking a course on Perturbation theory and as it's quite advanced the lecturer assumes everyone has a good level of maths. One of the parts is expanding roots of a quadratic equation about 0, I can understand how simple ones of the form $(1 + x)^2$ but I don't know where the answers are...
  8. 5

    Shortcut to taylor series of f, given taylor series of g

    So, I have the series of g(x) = e^{(x-1)^{2}} = 1 + (x-1)^{2} + \frac{(x-1)^{4}}{2} + \frac{(x-1)^{6}}{6} + ... + \frac{(x-1)^{2n}}{n!} and I am asked to find the series of f(x) = \frac{e^{(x-1)^{2}}-1}{(x-1)^{2}} for x \neq 1 and f(1) = 1. The Taylor series is centered about x = 1 I...
  9. K

    When does x=a in the taylor series stop being x=a?

    I'm having a hard time understanding the fundamentals of the taylor series. So I get how you continually take derivatives in order to find the coefficients but in order to do that we have to state that x=a. Well when we finally get done we have an infinite polynomial of...
  10. H

    Using Taylor expansion for limit solving

    Hi. I just want to ask: how can I realize that I need to do the 4th order taylor's expansion for solving a precise limit? e.g. \mathop {\lim }\limits_{x\to 0} \frac{{{e^x}-1-\frac{{{x^2}}}{2}+\sin x-2x}}{{1-\cos x-\frac{{{x^2}}}{2}}} We need the 4th order of the expansion but how can I realize...
  11. A

    Taylor and Euler Matlab Comparison for Numerical Analysis.

    1. Solve y'=3t^2y^2 on [0, 3] , y0 = −1, using Euler method and Taylor method of order 3. Compare your solutions to the exact solution. y(t)=(-1/((t^3)+1)) I DONT KNOW WHAT IS WRONG WITH MY PROGRAM! PLEASE HELP =D Homework Equations http://en.wikipedia.org/wiki/Euler_method...
  12. M

    Taylor expansion of an integral (for Thermodynamic Perturbations)

    In thermodynamic perturbation theory (chapter 32 in Landau's Statistical Physics) for the Gibbs (= canonical) distribution, we have E = E_0 + V, where V is the perturbation of our energy. When we want to calculate the free energy, we have: e^{-F/T} = \int e^{-(E+V)/T} \mathrm{d}\Gamma We can...
  13. J

    Approximating ln(x): Taylor Series Problem Solution

    Homework Statement The first three terms of a Taylor Series centered about 1 for ln(x) is given by: \frac{x^{3}}{3} - \frac{3x^{2}}{2} + 3x - \frac{11}{6} and that \int{ln(x)dx} = xlnx - x + c Show that an approximation of ln(x) is given by: \frac{x^3}{12} - \frac{x^2}{2} +...
  14. I

    Taylor expansion for f(x,y) about (x0,y0) ?

    Can someone please explain the Taylor expansion for f(x,y) about (x0,y0) ? Would really appreciate some sort of step by step answer :) thankyou
  15. R

    MHB Taylor and Geometric Series questions

    I've spent all day on this problem and am wasting precious time needed for other work - please give any input you can! The question: given two wages, w1 and w2 where w2 > w1... a. the difference between the wages as a proportion of the lower: a = (w2 - w1) / w2 b. the difference between the...
  16. D

    Finding a complex Taylor series

    Homework Statement Not much has gotten me in this class, and I almost want to say this has to be a typo, but I want someone else to check it out first. Homework question is that we need to show that cos(cos θ)*cosh(sin θ) = Ʃ(-1)ncos(nθ)/(2n)! for n>=0 There is a similar one involving...
  17. C

    MHB Pharaoh's Taylor series question from Yahoo Answers

    Part 1 of Pharaoh's Taylor series and modified Euler question from Yahoo Answers The Taylor series expansion about \(t=0\) is of the form: \(y(t)=y(0)+y'(0)t+\frac{y''(0)t^2}{2}+.. \)We are given \(y(0)\) and \(y'(0)\) in the initial condition, and so from the equation we have: \(y''(0) =...
  18. F

    Expanding f(x) = x/(x+1) about a=10

    Homework Statement Expand f(x) = x/(x+1) in a taylor series about a=10. Homework Equations f(x) = Ʃ (f^n(a)*(x-a)^n / n! The Attempt at a Solution I'm having a hard time arriving at the correct answer..I think I'm definitely getting lost somewhere along the way. Here's what I've...
  19. W

    Taylor Series: Can't quite work it out

    Hi Guys, Looking at some notes i have on conformal mapping and I have the following where z is complex and z* denotes its conjugate, R is a real number z* = -iR + R^2/(z-iR) and my lecturer says that using the taylor series we get, z* = -iR + iR(1+ z/iR + ...) I've been...
  20. S

    How is Taylor expansion used in physics?

    I wasn't sure where to put this, so I put this here! In the photo, you see there's written 'Taylor expanding for small delta-r2, we find' ... I really don't get the two steps in the next line. Any help would be greatly appreciated.
  21. I

    Taylor approximation of the Doppler Law for slow-moving emitters

    The relationship linking the emitted frequency Fe and the received frequency Fr is the Doppler Law: F_r = \sqrt \frac{1-\frac{v}{c}}{1-\frac{v}{c}} F_e The Taylor series for the function \sqrt\frac{1+x}{1-x} near x = 0 is 1+x+\frac{x^2}{2}+\frac{x^3}{3}+... On Earth, most objects travel...
  22. N

    Taylor polynom and some functionproblem.

    Anyone bored enough to want to help me out with some calculus? I got to deliver this in 6 hours and can't work these out. Help would be SO much appreciated, I've been at it all night and can't make it out. 1. y^2 - e^sin(x) + xy = sin (x)* cos (y) +3 assume y= y(x) and find y ' (0) 2...
  23. I

    Taylor series finding sin(x^2)+cos(x) from sin(x^2) and cos(x) alone

    If I want to find the taylor series at x = 0 for sin(x^2)+cos(x)... sin(x^2) = x^2 - x^6/3! + x^10/5! - x^14/7! ... cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! ... So why does sin(x^2) + cos(x) = 1 + x^2/2! + x^4/4! + 121x^6/6! ...? Thanks!
  24. S

    Taylor Series Remainder Theorem

    1. Prove that the MacLaurin series for cosx converges to cosx for all x. Homework Equations Ʃ(n=0 to infinity) ((-1)^n)(x^2n)/((2n)!) is the MacLaurin series for cosx |Rn(x)|\leqM*(|x|^(n+1))/((n+1)!) if |f^(n+1)(x)|\leqM lim(n->infinity)Rn=0 then a function is equal to its Taylor series...
  25. N

    Taylor Expansion: Do Assumptions Apply?

    Homework Statement Hi Say I want to Taylor-expand f(\omega + m\sin(\Omega t)) where ω and Ω are frequencies, m is some constant and t denotes time. Then I would get f(\omega + m\sin(\Omega t)) = f(\omega) + (m\sin(\Omega t)\frac{dI}{d\omega} + \ldots Is it necessary to make any...
  26. F

    Taylor expansion - imaginary coefficients?

    Homework Statement Find the first two non-zero terms in the Taylor expansion of \frac{x}{\sqrt{x^2-a^2}} where a is a real constantHomework Equations f(x)=f(x_0)+f^{\prime}(x_0)(x-x_0)+\frac{f^{\prime\prime}(x_0)}{2!}(x-x_0)^2+...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n The Attempt at a Solution If...
  27. C

    Advanced Calculus - Taylor errors

    This might be a weird quest and in the wrong section but does anyone know of a list with errors and corrections for the book "Advanced Calculus" by Taylor & Mann 3rd ed? Thanks
  28. C

    Inverse Laplace Transfrom - Taylor Series/Asymptotic Series?

    Homework Statement W(x,s)=(1/s)*(sinh(x*s^0.5))/(sinh(s^0.5)) Find the inverse laplace transform of W(x,s), i.e. find w(x,t). Answer: w(x,t)= x + Ʃ ((-1)^n)/n) * e^(-t*(n*pi)^2) * sin(n*pi*x) summing between from n=1 to ∞ Homework Equations An asymptotic series..?! The...
  29. Q

    What is the application of Taylor expansion in physics?

    i am very confuse how my profs always use taylor expansion in physics which somehow doesn't follow the general equation of f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)2 and so on... like for example, what is the taylor expansion of x - kx where k is small it was given as something like...
  30. T

    Finding the range of validity of a taylor series

    Homework Statement I have to give the range of validity for a Taylor series built from an expression of the form: (1+(a/b)x)^c Homework Equations The Attempt at a Solution Obviously the validity does not extend to x=-(b/a) on the negative side, but should I then state that...
  31. M

    Taylor expansions in two variables

    1. Problem: if f(1,3)=7, use Taylor expansion to describe f(1.2,3.1) and f(.9,2.8) if the partials of f are give by df/dx=.2 d^2f/dx^2=.6 df/dy=.4 d^2f/dy^2=.9 (you do not need to go beyond the second derivative for this problem) 2. I know from class how to do this if one variable changes...
  32. B

    Calculating Exponent Using Taylor Series To Given Precision

    Homework Statement The course is Computational Physics, but in a sense this is a pretty straight computer science or even mathematical challenge. The first part of the assignment - the relatively easy part - was to write a Fortran program to take two variables - the number to which e...
  33. S

    What did I do wrong here? (expressing root x as taylor series about a=4)

    Homework Statement Here is the question: I don't quite know what I did wrong. My method is below. Homework Equations The Attempt at a Solution f(x)=√x f'(x)=\frac{1}{2(x)^{1/2}} f''(x)=\frac{-1}{(2)(2)(x^{3/2}} a=4 f(a)=2 f'(a)=1/4...
  34. E

    Error estimate for Taylor polynomials

    Use the error estimate for Taylor polynomials to find an n such that | e - (1 + (1/1!) + (1/2!) + (1/3!) + ... + (1/n!) | < 0.000005 all i have right now is the individual components... f(x) = ex Tn (x) = 1/ (n-1)! k/(n-1)! |x-a|n+1 = 0.000005 a = 0 x = 1 I don't know where to go from here
  35. L

    Don't understand the taylor expansion?

    Not sure under which forum this should have gone under, anyway can someone who really understands it explain it to me in as simple terms as they can, from what I'm getting its approximates something for a function or something? No idea.
  36. O

    Taylor polynomial of degree 1 - solve for theta

    Homework Statement I was given the following problem, but I am having a hard time interpreting what some parts mean. We're given the equation sinθ+b(1+cos^2(θ)+cos(θ))=0 Assume that this equation defines θ as a function, θ(b), of b near (0,0). Computer the Taylor polynomial of...
  37. T

    Derive using Taylor series/Establish error term

    Homework Statement Derive the following formula using Taylor series and then establish the error terms for each. Homework Equations f ' (x) ≈ (1/2*h) [4*f(x + h) - 3*f(x) - f(x+2h)] The Attempt at a Solution I honestly have no idea how to go about deriving this. The professor did...
  38. W

    Predictor-corrector, starting values with Taylor method

    Homework Statement Hi! I need to solve the following ODE: xy'=1-y+x^2y^2, \qquad y(0)=1 using a predictor-corrector method. Starting values need to be found using a Taylor method. The exact solution is of the form \frac{\tan{x}}{x} Homework Equations Taylor method of third order...
  39. D

    MHB Expanding Taylor Series to Get Approximate Answer

    $1+v_{t+1} = (1+v_t)\exp\left(-rv_{t-1}\right)\approx (1+v_t)(1-rv_{t-1})$ The book is linearizing the model where we generally use a Taylor Series. How was the expression expanded in the Taylor Series to get the approximate answer? Thanks.
  40. T

    Calculus - Taylor Expansion, maybe. Not sure how to simplify.

    I am attempting to complete a problem for a problem set and am having difficulty simplifying an expression; any help would be greatly appreciated! The question is a physics question which attempts to derive an equation for the temperature within a planet as a function of depth assuming...
  41. A

    What is the simplification of the second order Taylor expansion for F(x+h)?

    Homework Statement Show that if F is twice continuously differentiable on (a,b), then one can write F(x+h) = F(x) + h F'(x) + \frac{h^2}{2} F''(x) + h^2 \varphi(h), where \varphi(h) \to 0 as h\to 0. Homework Equations The Attempt at a Solution I'm posting this here...
  42. C

    Taylor series for cos[1/(1-z^2)]

    Bit stuck on this. I tried writing 1/(1-z^2) as taylor series then Cos z as taylor series, then substituting one into the other but it looked a bit dodgy. Can one simple substitute like this?
  43. C

    Quicker Method to Find Taylor Series of sinz - sinhz?

    I have to find the first three non zero terms of this series by hand. I know the answer and it is -(z^3/3) - z^7/2520 - z^11/19958400 Which will take ages to get to by brute force. Is there a quicker way?
  44. D

    MHB Finding Taylor Series of $\dfrac{1}{z-i} \div \left(z+i\right)$

    I am trying to find the Taylor series for $$\displaystyle \dfrac{\left(\dfrac{1}{z-i}\right)}{z+i} $$ where z is a complex number.There is a reason it is set up as a fraction over the denominator so let's not move it down.
  45. R

    Alt. approach to Taylor series of derivative of arcsin(x)?

    Hi there, I was hammering out the coefficients for the Taylor Series expansion of f(x) = \frac{1}{\sqrt{1-x^2}}, which proved to be quite unsatisfying, so decide to have a look around online for alt. approaches. What I found (in addition to the method that uses the binomial theorem) was...
  46. P

    Taylor series expansion for gravitational force

    Homework Statement The magnitude of the gravitational force exerted by the Earth on an object of mass m at the Earth's surface is Fg = G*M*m/ R^2 where M and R are the mass and radius of the Earth. Let's say the object is instead a height y << R above the surface of the Earth. Using a...
  47. S

    Hessian matrix in taylor expansion help

    Homework Statement Find the critical point(s) of this function and determine if the function has a maxi- mum/minimum/neither at the critical point(s) (semi colons start a new row in the matrix) f(x,y,z) = 1/2 [ x y z ] [3 1 0; 1 4 -1; 0 -1 2] [x;y;z] Homework Equations The...
  48. N

    Integration of O() terms of the Taylor series

    Hello, I have two functions say f1(β) and f2(β) as follows: f1(β)=1/(aδ^2) + 1/(bδ) + O(1) ... (1) and f2(β)= c+dδ+O(δ^2) ... (2) where δ = β-η and a,b,c,d and η are constants. Eq. (1) and (2) are the Taylor series expansions of f1(β) and f2(β) about η...
  49. A

    (Deceptively?) Simple question about Taylor series expansions

    Under what circumstances is it correct to say of the function u(x) \in L^2(-\infty,\infty) that u(x-t) = u(x) - \frac{du}{dx}t + \frac 12 \frac{d^2u}{dx^2}t^2 - \cdots = \sum_{n=0}^\infty \frac{u^{(n)}(x)}{n!}(-t)^n.
  50. T

    Multivariate Taylor expansion or else a double integral identity

    Homework Statement This is part of a larger problem, but in order to take what I believe is the first step, I need to take the Taylor series expansion of f(x,y) = \cos\sqrt{x+y} about (x,y) = (0,0) On the other hand, the purpose of doing this expansion is to find an asymptotic expression for...
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