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. 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...
  2. 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!
  3. 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'}...
  4. 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...
  5. 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...
  6. S

    How do I prove that both are equivalent limits

    Homework Statement If k is a positive integer, then show that ##\lim_{x\to\infty} (1+\frac{k}{x})^x = \lim_{x\to 0} (1+kx)^\frac{1}{x}## Homework Equations L'Hopitals rule, Taylor's expansion The Attempt at a Solution How should I begin? Should I prove that both has the same limit, or is...
  7. 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...
  8. 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.
  9. T

    Taylor Expansion to Understanding the Chain Rule

    I don't understand this as isn't according to chain rule, . So where is the in the above derivative of F(t)? Source: http://www.math.ubc.ca/~feldman/m226/taylor2d.pdf
  10. noowutah

    Stationary point for convex difference measure

    Let \mathbb{S}^n be a simplex in \mathbb{R}^{n+1}, so \mathbb{S}^{n}=\{x\in\mathbb{R}^{n+1}|\sum{}x_{i}=1\}. Let D be a difference measure on \mathbb{S}^{n} with D(x,x)=0 and x=y for D(x,y)=0. D is also smooth, so differentiable as much as we need. Let (R) be a convexity requirement for D...
  11. S

    Taylor expansion of the square of the distance function

    Does it make a sense to define the Taylor expansion of the square of the distance function? If so, how can one compute its coefficients? I simply thought that the square of the distance function is a scalar function, so I think that one can write $$ d^2(x,x_0)=d^2(x'+(x-x'),x_0)=d^2(x',x_0) +...
  12. R

    (Thermo) Energy as Taylor expansion

    Homework Statement [/B] I've attached a screenshot of the problem, which will probably provide much better context than my retelling. I'm having problems with parts f and g. The most relevant piece of information is: "To get used to the process of Taylor expansions in two variables, first we...
  13. S

    Riemannin generalization of the Taylor expansion

    I thought about the Taylor expansion on a Riemannian manifold and guess the Taylor expansion of ##f## around point ##x=x_0## on the Riemannian manifold ##(M,g)## should be something similar to: f(x) = f(x_0) +(x^\mu - x_0^\mu) \partial_\mu f(x)|_{x=x_0} + \frac{1}{2} (x^\mu - x_0^\mu) (x^\nu -...
  14. I

    Finding the Limit of a Sequence with Taylor Expansion and Exponentials

    Homework Statement Let ##x_n## be the solution to the equation ##\left( 1+\frac{1}{n} \right)^{n+x} = e## Calculate ##\lim_{n\to \infty} x_n## Homework Equations N/A The Attempt at a Solution Since ##\lim_{n \to \infty} \left(1+ \frac{1}{n} \right) = e## that tells me that ##\lim_{n\to...
  15. T

    Expanding a function in terms of a vector

    Homework Statement ## L (v^2 + 2 \pmb{v} \cdot \pmb{ \epsilon } ~ + \pmb{ \epsilon} ^2)##, where ## \pmb{\epsilon}## is infinitesimal and ##\pmb{v}## is a constant vector (## v^2 ## here means ## \pmb{v} \cdot \pmb{v} ## ), must be expanded in terms of powers of ## \pmb{\epsilon} ## to give...
  16. J

    Very long Taylor expansion/partial fraction decomposition

    Homework Statement I want to express the following expression in its Taylor expansion about x = 0: $$ F(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)} $$ The Attempt at a Solution First I tried to rewrite the function in partial fractions (its been quite a while since I've last...
  17. nuuskur

    Taylor Series for Square Root Function

    Homework Statement Expand ##f(x) = \sqrt{2x+1}## into a Taylor series around point ##c=1##. Find the interval of convergence. Homework EquationsThe Attempt at a Solution I do know that ##f(x) = \sum\frac{1}{n!}f^{(n)}(c)(x-c)^n## assuming the function is representable as a Taylor series. How...
  18. M

    MHB Taylor expansion of second order

    Hey! :o I have to find the Taylor expansion of second order of the following functions with center the given point $(x_0, y_0)$. $f(x, y)=(x+y)^2, x_0=0, y_0=0$ $f(x, y)=e^{-x^2-y^2}\cos (xy), x_0=0, y_0=0$ I have done the following: The Taylor expansion of second order of $f...
  19. D

    How do I write taylor expansion as exponential function?

    How do I write taylor expansion of a function of x,y,z (not at origin) as an exponential function? Please see the attached image. I need help with the cross terms. I don't know how to include them in the exponential function?
  20. F

    Unitary translation operator and taylor expansion

    Homework Statement I have quite a straightforward question on the taylor expansion however I will try to provide as much context to the problem as possible: ##T(a)## is unitary such that ##T(-a) = T(a)^{-1} = T(a)^{\dagger}## and operates on states in the position basis as ##T(a)|x\rangle =...
  21. M

    MHB Taylor Expansion: Wondering Which is Right?

    Hey! :o I want to find the taylor expansion of $f(x, y)=x^2 (3y-2x^2)-y^2 (1-y)^2$ at the point $(0, 1)$ and I got the following: $$f(x, y)=3x^2-(y-1)^2+3x^2 (y-1)-2 (y-1)^3-2x^4-(y-1)^4$$ but a friend of mine got the following result: $$f(x, y)=3x^2-(y-1)^2+3x^2 (y-1)+3x...
  22. V

    How to Apply Taylor Series at Infinity for x >> 1?

    Homework Statement How to use Taylor series for condition x>>1? For example f(x)=x\sqrt{1+x^2}(2x^2/3-1)+\ln{(x+\sqrt{1+x^2})} Homework EquationsThe Attempt at a Solution I try to derived it and limit to infinity...for example first term \frac{x^4}{3\sqrt{1+x^2}}. Limit this to infinity is...
  23. I

    Relativistic mechanics Taylor expansion

    Homework Statement For a particle traveling near the speed of light, find the first non-vanishing term in the expansion of the relative difference between the speed of the particle and the speed of light, (c-v)/c, in the limit of very large momentum p>>mc. Hint: Use (mc/p) as a small parameter...
  24. skate_nerd

    MHB Does this problem warrant a taylor expansion? (solid state physics)

    The whole problem I'm doing here is not even really relevant, so I won't go too much into it...I'm told to find an atomic form factor given some certain conditions, and I do a big gross integral and got this: $$f=(\frac{4}{4+(a_oG)^2})^2$$ where \(a_o\) is the Bohr radius and \(G\) is the...
  25. V

    Proving c_{n} = 0 Using Taylor Expansion: Homework Help

    Homework Statement Suppose that f(x)=\sum_{n=0}^{\infty}c_{n}x^{n}for all x. If \sum_{n=0}^{\infty}c_{n}x^{n} = 0, show that c_{n} = 0 for all n. Homework Equations The Attempt at a Solution I know, by using taylor expansion, c_{n}=\frac{f^{n}(0)}{n!}, and because...
  26. T

    You can edit the title of the first post and add [solved] at the beginning.

    I'm confused by problem 2.31 in mathematical tools for physics. Problem: 2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The Doppler effect for light, including relativistic effects is different still. Show that for low speeds they are...
  27. KiNGGeexD

    Using Taylor expansion to find the limits of a function

    https://www.physicsforums.com/attachments/68247 I had been assigned this problem, I worked out the expansions (for practice) so they could have errors in them! I got to a point (in the photograph) where I could take out a common factor of 1/x but I'm pretty stumped although via other methods...
  28. T

    One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone

    Hi There, I came across the following passage in Sam Glasstone's 'Nuclear Reactor Engineering' See where I underlined in red that taylor series expansion? I don't understand how (dt/dx)_(x+dx) is equal to that. I know it's a Taylor series expansion, but where did the x+dx go?
  29. K

    Derive the analytic expression of a function by its Taylor expansion

    Homework Statement Actually this is not from homework. It occurs in my brain this afternoon. Is it possible to derive the analytic expression of a function by its Taylor series expansion? For example, given the following expansion, how to derive the analytic expression of it? f(x) =...
  30. U

    MHB Manipulating Taylor Expansion for Sample Mean, Variance, Skewness & Kurtosis

    I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n x_i^4 + \cdots \right)$$ Now let $$Y = \frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n}...
  31. N

    Higher order derivatives with help of Taylor expansion?

    Homework Statement Function f(x) = x^2/(x-1) should be expanded by Taylor method around point x=2 and 17th order derivative at that point should be calculated. Homework Equations Taylor formula: f(x)=f(x0)+f'(x0)*(x-x0)+f''(x0)*(x-x0)^2+... The Attempt at a Solution I...
  32. U

    Taylor Expansion on Determinant

    Homework Statement Show by direct expansion that: det (I + εA) = 1 + εTr(A) + O(ε2) Homework Equations f(x) = f(a) + (x-a)f'(a) + (1/2)(x-a)2f''(a) + ... The Attempt at a Solution Does the question mean Taylor expansion when they say 'direct expansion'? I'm kind of stuck on...
  33. Z

    How Do You Calculate the Taylor Expansion of e^(2-x)?

    Homework Statement Find e^{2-x} using taylor/mclaurin expansionHomework Equations e^1 = \sum_{n=0}^\inf \frac{1}{n!} e^x = \sum_{n=0}^\inf \frac{x^n}{n!} The Attempt at a Solution Can I just do: e^{1+1-x} [\sum_{n=0}^\inf \frac{1}{n!}*\sum_{n=0}^\inf \frac{1}{n!} *...
  34. R

    Taylor Expansion of f+df About x?

    How would one expand f+df about x? I'm messing something up in the process and can't seem to resolve it lol
  35. mnb96

    Question on Taylor expansion of first order

    Hello, according to my textbook, the Taylor expansion of first order of a scalar function f(t) having continuous 2nd order derivative is supposed be: f(t) = f(0) + f'(0)t + \frac{1}{2}f''(t^*)t^2 for some t^* such that 0\leq t^* \leq 1 Quite frankly, I have never seen such a formulation...
  36. D

    Determinants and taylor expansion

    I'm doing a proof, and near the last step I want to write the expression, \frac{d}{dt} \det{A(t)} = \lim_{\epsilon \to 0} \frac{\det{(A+\epsilon \frac{dA}{dt})} - \det{A}}{\epsilon} which produces the right answer, so I believe that it may be correct. This looks very much like a Taylor...
  37. Fernando Revilla

    MHB Verification of Limit Using Taylor Expansion: x-ln(1+x)/x^2 = 1/2

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  38. DeusAbscondus

    MHB Taylor expansion to express e^x

    Hi folks, If $e^x= \Sigma_{k=0}^\infty \frac{x^k}{k!}$ what do I evaluate $x$ at? How does the sigma notation tell me what to do with $x$? $$e^x= \Sigma_{k=0}^\infty \frac{x^k}{k!}\ = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} ... \text {ad infinitum}$$ Sorry, I just realized my error...
  39. A

    Taylor expansion at infinity of x/1+e^(1/x)

    I have some problems finding Taylor's expansion at infinity of f(x) = \frac{x}{1+e^{\frac{1}{x}}} I tried to find Taylor's expansion at 0 of : g(u) = \frac{1}{u} \cdot \frac{1}{1+e^u} \hspace{10 mm} \mbox{ where } \hspace{10 mm} u = 1/x in order to then use the known expansion of...
  40. fluidistic

    Taylor expansion, thermodynamics

    Homework Statement I'm asked to show that the relation ##S(U+\Delta U, V+ \Delta V , n )+S(U-\Delta U, V- \Delta V , n ) \leq 2 S(U,V,n)## implies that ##\frac{\partial ^2 S}{\partial U ^2 } \frac{\partial ^2 S}{\partial V ^2}- \left ( \frac{\partial ^2 S }{\partial U \partial V} \right ) \geq...
  41. P

    Taylor Expansion where the derivatives are undefined?

    Homework Statement Expand x/(x-1) at a=1 The book already gives the expansion but it doesn't explain the process. The expansion it gives is: x/(x-1) = (1+x-1)/(x-1) = (x-1)^(-1) + 1 Homework Equations The Attempt at a Solution I've already solved for the Mclaurin expansion for the same...
  42. O

    Sakurai page 54: Is this a Taylor expansion?

    From page 54 of 'Modern Quantum Mechanics, revised edition" by J. J. Sakurai. Obtaining equation (1.7.15), \begin{eqnarray} \left(1- \frac{ip\Delta x'}{\hbar} \right) \mid \alpha \rangle & = & \int dx' \mathcal{T} ( \Delta x' ) \mid x' \rangle \langle x' \mid \alpha \rangle \\ & = & \int...
  43. C

    Non-relativistic limit of Compton scattering -Taylor expansion?

    Homework Statement In the non-relativistic limit, the equation for Compton scattering is v' / v = 1 / (1+x) where x = (hv/mc2)(1-Ω.Ω') Show that the change in frequency Δv = v' - v is given by Δv / v = -x Homework Equations The Attempt at a Solution I rearranged the...
  44. X

    Taylor expansion of a vector function

    Could someone please explain how does this taylor expansion work: 1/|r-r'| ≈ 1/r+(r.r')/r3 possibly you have to taylor expand twice to get this result, an attempt at which led me nowhere, surely it cannot be this complicated. any useful comment about this would be greatly appreciated...
  45. B

    Taylor expansion for matrix logarithm

    A paper I'm reading states the that: for positive hermitian matrices A and B, the Taylor expansion of \log(A+tB) at t=0 is \log(A+tB)=\log(A) + t\int_0^\infty \frac{1}{B+zI}A \frac{1}{B+zI} dz + \mathcal{O}(t^2). However, there is no source or proof given, and I cannot seem to find a...
  46. L

    How was the Taylor expansion for SSB in superconductors done?

    I am reading about spontaneous symmtry breaking for superconductors and came a cross to this simple statement: Here is the potential for complex scalar field: V = 1/2 \lambda^2 (|\phi|^2 -\eta^2)^2 . Scalar field is small and we can expand its modulus around \eta : \phi(x) =...
  47. M

    Taylor expansion of an electrostatics problem

    Homework Statement The problem has six charges that are at the corners of a regular hexagon in the xy plane, each charge a distance a from the origin. I have already solved for the electric fields in the x and y direction and now am trying to apply an approximation for the field on the x-axis...
  48. Vola

    How to Convert Taylor Expansion to Summation Notation and Vice Versa?

    Hi everyone, Is there a certain technique or a program for converting Taylor expansion to summation notation form and vice versa. Thank you in advance.
  49. G

    Taylor Expansion of Natural Logarithm

    Hello! I was trying to look for a possible expansion of the ln function. The problem is, that there is no expansion that can be used in all points (like there is for e, sine, cosine, etc..) Why do you think that is? To clarify: Let's say i do the MacLaurin expansion of ln(x+1)...
  50. J

    Taylor Expansion of A Magnetic Field

    Quick question about Taylor expansions that I am getting pretty confused about. Let's say using biot savart I want to find the field of a INFINITE helix (http://en.wikipedia.org/wiki/Helix) along the axis but with very slight displacements of x and y (x+ε,y+ε). Here is a series of steps I will...
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