What is Taylor expansion: Definition and 174 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. T

    Confused about Taylor Expansion Strategy

    In many of my physics classes we have been using Taylor Expansions, and sometimes I get a bit confused. For example, I feel like different things are going on when one expands (1-x)^-2 vs. e^(-Ax^2), where I just have some constant in front of x^2 to help make my point. To keep things simple...
  2. fluidistic

    I don't understand a step in my notes, about Taylor expansion

    Homework Statement P_0 (t+dt)=P_0(t)(1-\gamma dt ) (1) Therefore P_0 (t)+\frac{dP_0 (t)}{dt} \approx P_0 (t)-\gamma P_0(t)dt. (2) Where the approximation is due to a Taylor expansion apparently. Homework Equations Taylor expansion of f around x_0 : f(x)\approx...
  3. T

    Taylor expansion of a vector field

    I was wondering if such an approximation is possible and plausible... The first term would have to look sth like this: \vec{f}(\vec{x_{0}}) + \textbf{J}_{\vec{f}}(\vec{x_{0}})\cdot(\vec{x}-\vec{x_{0}}) No clue about the second term though... We would have to calculate the Jacobian of the...
  4. R

    Mathematica Taylor Expansion in Mathematica (Multivariate)

    Hi! I'm trying to linearize a function f which is dependent of 4 variables, each one dependent of time. f[var1_,var2_,var3_,var4_]:= ... expression I use Series[f, {var1,var10 ,1},{var2,var20 ,1},...] syntax as I read on the documentation center. The problem is that the program...
  5. H

    Multivariable Calculus book with Taylor expansion for several variables

    Hi. I borrowed many multivariable Calculus book so that I can choose one for the next semester. The one I liked most is Multivariable Calculus by Ron Larson. It is full of graphics and colours, somthing that is essential to understand functions of two variables. The only thing is that it does...
  6. U

    Weighted Sum in Taylor Expansion (Partial Derivatives)

    Homework Statement From step 1 to step 2, what do they mean by "Taking the weighted sum of the two squares " ? I tried and expanded everything in step 2 and it ends up as the same as step 1 (as expected), The Attempt at a Solution I tried looking up "weighted sum" and "...
  7. B

    Variational Operator/First Variation - Taylor Expansion

    Homework Statement Folks, how is the following expansion obtained for the following function ##F(x,u,u')## where x is the independent variable. The change ##\epsilon v## in ##u## where ##\epsilon## is a constant and ##v## is a function is called the variation of ##u## and denoted by...
  8. Rococo

    Trying to expand (1+x)^n using Taylor Expansion

    Homework Statement I need to use Taylor Expansion to show that: (1+x)^n = 1 + nx + n(n-1)(x^2)/2! + ...Homework Equations y(x0 + dx) = y(x0) + dx(dy/dx) + [(dx)^2/2!](d^2y/dx^2) + ... The Attempt at a Solution I've only just begun Taylor Expansion, according to my textbook I need the above...
  9. T

    Where does this method of constructing the taylor expansion of arctanx fails?

    Homework Statement arctan x = ∫du/(1+u2), from 0 to x Homework Equations The Attempt at a Solution I noticed that 1/(1+u2) = 1/(1+u2)1/2 × 1/(1+u2)1/2. I decided to take the taylor series expansion of 1/(1+u2)1/2, square the result and then integrate. I got 1/(1+u2)1/2 =...
  10. 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?
  11. 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...
  12. 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...
  13. 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...
  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. 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.
  16. 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...
  17. 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...
  18. 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...
  19. 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.
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. N

    Lagrangian Taylor expansion

    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
  25. 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) +...
  26. 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...
  27. 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...
  28. 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...
  29. Y

    Please explain Taylor expansion in radiation.

    For retarded scalar potential of arbigtrary source around origin: V(\vec r, t) = \frac 1 {4\pi\epsilon_0}\int \frac { \rho(\vec r\;',t-\frac {\eta}{c}) }{\eta} d\;\tau' \;\hbox { where }\;\eta =\sqrt{r^2 + r'^2 - 2 \vec r \cdot \vec r\;' } Where \;\vec r \; point to the field point where V...
  30. P

    Inverse Laplace transform for small 's', Taylor expansion

    Dear all, This question is close to the post "Laplace transform of a Taylor series expansion" in PhysicsForums.com, dated Jul06-09. This is my problem: Consider the Laplace transform F(s) = 1 / ( s - K(s) ) , where K(s) = -1/2 + i/(2*Pi) * ln[ ( Lambda - (b+i*s) )/( b + i*s...
  31. T

    How did Jackson simplify the Taylor expansion for charge density in 3-D?

    On p35 of Jackson's Classical Electrodynamics 3rd Edition, the author gives the expansion of the charge density \rho(\mathbf{x'}) around \mathbf{x'}=\mathbf{x} as \rho(\mathbf{x'}) = \rho(\mathbf{x}) + \frac{r^2}{6}\nabla^2\rho + ... where r = |\mathbf{x} - \mathbf{x'}| My question is...
  32. J

    A little help with this Taylor expansion please?

    I have this translation operator T(a) that acts on a function y(x) and causes the transformation T(a)y(x) = y(x+a). I am supposed to be "expanding y(x+a) as a taylor series in a" to show that T(a)=eipa, where p is the operator p = -i.d/dx] So, I've started out with the general equation for the...
  33. B

    Understanding Taylor Expansions of Gradients in Vector Calculus

    What does it mean to have a taylor expansion of a gradient (vector) about the position x? I.e. taylor expansion of g(x + d) where g is the gradient and d is the small neighborhood.
  34. E

    How Does Adding a Hydrogen Layer Affect Tunneling Coefficients in STM Analysis?

    Homework Statement In my never ending quest to suck and never be able to do Taylor Expansions, I have another one. I hope one day I'll be able to do these. I have an unknown material and a scanning tunnel microscope. A layer of hydrogen atoms of radius R are added to the surface. This of...
  35. L

    Help understanding Taylor expansion etc.

    Well - I am a 16 year old student, whos really interested in math. I do a lot of studying on my own, because I am a bit bored with the present math in school. Right now I am reading about solving differential equations with power series. I can do this, and i do understand the recurrence...
  36. N

    How can I simplify this Taylor expansion?

    I want to show this taylor expansion: \frac{1}{\sqrt{1+{x}^{2}}} \rightarrow x^2 what I keep getting is something to the x^3 could some one please help me with this simple expansion?
  37. Q

    The integral of the taylor expansion of any expression

    ok .lets say the expression we have is ex the taylor expansion becomes 1+x+x2/2+... integrating becomes x+x2/2+x3/6+...+c so how do we know that c = 1? for it to become back to ex becos it is said that integral of ex = ex do we just let x be 0 to find c = 1? does it work for all...
  38. B

    Jackson - p. 35 - taylor expansion of charge density

    Homework Statement With a Taylor series expansion of the well-behaved \rho ({\bf{x'}}) around {\bf{x'}} = {\bf{x}}, one finds the Taylor expansion of the charge density to be, \rho ({\bf{x'}}) = \rho ({\bf{x}}) + {\textstyle{1 \over 6}}{r^2}{\nabla ^2}\rho + ... Homework Equations...
  39. I

    Taylor Expansion for (a(1+z)^3 + b)^-1/2 around z=0 to First Order

    Homework Statement I have the function (a(1+z)3 + b)-1/2 and i need to taylor expand it around z=0 to the first order, a and b are constants, there sum is equal to one. I have the answer: 1 - (1+q)z where q = a/2 - b This is in my physics book but it does not explain the...
  40. L

    Fractional Uncertainty in Total Mass fM: Taylor Expansion & Relation to fd

    Homework Statement If fd is the fractional uncertainty on the distance, what is the fractional uncertainty on the total mass fM? Hint: use our Taylor expansion approximation that (1 +- x) ^a ~ 1 +- ax when x << 1. The fractional uncertainty in the mass fM will be related to fd in a simple...
  41. D

    What is the Taylor polynomial for x^x around the point a=1?

    hello, please help to calculate the taylor polynomial for http://latex.codecogs.com/gif.latex?f(x)=x^{x}-1 around the point a=1 i thought to write it as g(x)=x^x and then f(x)=g(x)-1 and then find the polynomial for g(x) as lng(x)=xln(x) but it seems incorrect.
  42. S

    How is the Taylor expansion of 1/|r-r'| done in electrodynamics?

    I am wroking through an electrodynamics textbook and there is this Taylor expansion to do later a multipole expansion. But I can't figure out how the author does it. Please any help? the expansion: \frac{1}{|\vec{r}-\vec{r'}|} = \frac{1}{r} - \sum^3_{i=1} x'_i \frac{\partial}{\partial...
  43. T

    Proving Existence of a Point c for Taylor Expansion | Homework Statement

    Homework Statement Let f be differentiable on [a,b] and f'(a)=f'(b)=0. Prove that if f'' exists then there exists a point c in (a,b) such that test |f''(c)| \geq \frac{4}{(b-a)^2}|f(b)-f(a)| Homework Equations All of the equations are supposed to be in absolute value but I had...
  44. Q

    Finding Taylor Expansion of f(x) and ln(1+x^2)

    Homework Statement a) Using a geometric series, find the Taylor expansion of the function f(x) = x/(1+x^2) b) Use the series found in (a) to obtain the Taylor expansion of ln(1 + x^2) Homework Equations The Attempt at a Solution I really don't know where to start; I can't find...
  45. N

    Taylor expansion of gaussian integral with respect to variance

    Hi everyone. The problem I have to face is to perform a taylor series expansion of the integral \int_{-\infty}^{\infty}\frac{e^{-\sum_{i}\frac{x_{i}^{2}}{2\epsilon}}}{\sqrt{2\pi\epsilon}^{N}}\cdot e^{f(\{x\})}dx_{i}\ldots dx_{N} with respect to variance \epsilon. I find some difficulties...
  46. R

    Estimating (16.1)1/4 using Taylor's Expansion at x=16

    Homework Statement Use the taylor's expansion of f(x)= x1/4 about x= 16 to estimate (16.1)1/4 Homework Equations Taylors formula: f(a) + f'(a) (x-a) + (f''(a)/2!) (x-a)2+...The Attempt at a Solution Ok I have calculate the taylor expansion to be: 2 + (1/32) (x-16)-(3/320) (x-16)2+ (7/262144)...
  47. K

    Taylor Expansion for rational function

    Homework Statement Find the taylor expansion of the following formula in the case where r > > d to the first order in \epsilon = \frac{d}{r} \frac{1}{r_{+}} = \frac{1}{\sqrt{r^{2} + (\frac{d}{2})^{2} - rdcos\theta}} Homework Equations (1 + \epsilon)^{m} = 1+m\epsilon, where...
  48. O

    Calc 2 - Taylor Expansion Series of x^(1/2)

    Homework Statement f(x) = \sqrt{x}, a = 4 Homework Equations f(x) = \sumf^{n}(a)/n! (x-a)^{n} The Attempt at a Solution f(x) = x^{1/2} f^{'}(x) = \frac{1}{2}x^{1/2} f^{2}(x) = -\frac{1}{2}*\frac{1}{2}x^{-3/2} f^{3}(x) = \frac{1}{2}*\frac{1}{2}*\frac{3}{2}x^{-5/2} f^{4}(x) =...
  49. C

    Mathematica How to compute a Taylor expansion for f(x,y) using Mathematica?

    Can someone pls explain hot to compute a taylor expansion for f(x,y) using mathematica
  50. C

    Taylor Expansion of 1/(r-r'): Explained

    Homework Statement Could someone please explain how the taylor expansion of 1/(r-r') turns into ( 1/r+(r'.r)/r^3 + (3(r.r')^2-r^2r'^2)/2r^5 +...) Homework Equations The Attempt at a Solution
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