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. A

    I Taylor Series for Potential in Crystals

    Hi, I've been reading the passage attached below and from what I understand we are looking at a 1D chain of atoms and if anyone atom moves it changes the potential for surrounding atoms and cause a change in energy in the system so the total energy is dependent on all the positions of the atoms...
  2. W

    I 1D scattering: Taylor expansion

    Hi all, I'm having a problem understanding a step in an arxiv paper (https://arxiv.org/pdf/0808.3566.pdf) and would like a bit of help. In equation (29) the authors have $$R = \frac{\sigma}{\sqrt{\pi}} \int dk \ e^{-(k - k_0)^2 \sigma^2} \ \Big( \frac{ k - \kappa}{ k+ \kappa} \Big)^2$$ where...
  3. S

    How to Evaluate the 8th Derivative of a Taylor Series at x=4

    Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## Homework Equations The Taylor Series Equation The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
  4. SebastianRM

    I What is the tangential component? Taylor p.347

    I do not understand why the tan component for a gravity affected by the centrifugal force: g = Ω^2 * R * sinθ * cosθ So I tried to draw this: using a "big" X-shaped axis where the / component goes along the main gravity direction while \ points normal to / this direction. Then the centrifugal...
  5. SpaceIsCool

    Transverse velocity and real/imaginary parts?

    Homework Statement The transverse velocity of the particle in Sections 2.5 and 2.7 is contained in (2.77), since By taking the real and imaginary parts, find expressions for v_x and v_y separately. Based on these expressions describe the time dependence of the transverse velocity. Homework...
  6. SpaceIsCool

    Can This Differential Equation Be Solved by Separation of Variables?

    Homework Statement We solved the differential equation (2.29), , for the velocity of an object falling through air, by inspection---a most respectable way of solving differential equations. Nevertheless, one would sometimes like a more systematic method, and here is one. Rewrite the equation...
  7. Zack K

    Approximating the force on a dipole Taylor series

    Homework Statement Show that the magnitude of the net force exerted on one dipole by the other dipole is given approximately by:$$F_{net}≈\frac {6q^2s^2k} {r^4}$$ for ##r\gg s##, where r is the distance from one dipole to the other dipole, s is the distance across one dipole. (Both dipoles are...
  8. N

    B "Spacetime Physics" by Taylor and Wheeler

    I've tried to get through this book a couple times in the past but didn't succeed, and am now trying it a third time. I managed to re-read and redo the problems from the first chapter recently, and now I'm on the second one where the authors get into free float frames and the force of gravity...
  9. L

    MHB Using Standard Taylor Series to build other Taylor Series

    Hello there, I am studying Taylor series, and in the slides given to us we calculated the taylor series of ln $(\frac{1+x}{1-x} )$ = ln(1 + x) − ln(1 − x), by using standard Taylor series of ln(1 + x). The notes then proceed to say : " It can be shown that every positive real number t can be...
  10. K

    Limit of Taylor Polynomial for Tn(x) as n Approaches Infinity

    Homework Statement Let Tn(x)=1+2x+3x^2+...+nx^(n-1) Find the value of the limit lim n->infinity Tn(1/8).The Attempt at a Solution How do I solve this? I know how to write the polynomial as a series, but not sure how if this is the best way of finding the limit.
  11. K

    Taylor polynomial, approximative solution of this equation

    Homework Statement The equation 4x = (1/3)*cos(3x) has a solution on the interval [0,1]. Find an approximative solution by replacing the right hand side with a Taylor polynomial of degree 2 around 0. Homework EquationsThe Attempt at a Solution So as I understand the task we should find a...
  12. S

    B GramSchmidt process for Taylor basis

    Why are the limits as so for the integral?
  13. U

    A Taylor expansion for a nonlinear system and Picard Iterations

    Hello guys I struggle since yesterday with the following problem I am reading the book "Elements of applied bifurcation theory" by Kuznetsov . At one point he has the following Taylor expansion of a nonlinear system with respect to x=0 where ##x\in \mathbb(R)^n## $$\dot{x} = f(x) = \Lambda x +...
  14. Jozefina Gramatikova

    By calculating a Taylor approximation, determine K

    Homework Statement Homework Equations [/B]The Attempt at a Solution Can somebody explain to me how did we find the function in red? Thanks
  15. C

    A Questions about the energy of a wave as a Taylor series

    I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = ao + a1 A + a2A2 ... Also, that the energy doesn't depend on phase, so only even terms will...
  16. Mr Davis 97

    I Different series than Taylor series for a function

    I am trying to solve an integral that has ##\frac{1}{1+x^2}## as a factor in the integrand. In my book it is claimed that if we use ##\displaystyle \frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-1)^n \frac{1}{x^{2n+2}}## the problem can be solved immediately. But, I am confused as to where this series...
  17. R

    What is the Taylor expansion of x/sin(ax)?

    Hey everyone 1. Homework Statement I want to compute the Taylor expansion (the first four terms) of $$f(x) =x/sin(ax)$$ around $$x_0 = 0$$. I am working in the space of complex numbers here. Homework Equations function: $$f(x) = \frac{x}{\sin (ax)}$$ Taylor expansion: $$ f(x) = \sum...
  18. N

    How can the Taylor expansion of x^x at x=1 be simplified to make solving easier?

    Homework Statement Find the Taylor expansion up to four order of x^x around x=1. Homework EquationsThe Attempt at a Solution I first tried doing this by brute force (evaluating f(1), f'(1), f''(1), etc.), but this become too cumbersome after the first derivative. I then tried writing: $$x^x =...
  19. Ben Geoffrey

    I Taylor Series Expansion of Quadratic Derivatives: Goldstein Ch. 6, Pg. 240

    Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below
  20. C

    News How Did Nobel Laureate Richard Taylor Impact Physics?

    I noticed this today: Nobel Prize-winning physicist Richard Taylor dies at 88 RIP
  21. M

    I Which x_0 to use in a Taylor series expansion?

    I already learn to use Taylor series as: f(x) = ∑ fn(x0) / n! (x-x0)n But i don´t see why the serie change when we use differents x0 points. Por example: f(x) = x2 to express Taylor series in x0 = 0 f(x) = f(0) + f(0) (x-0) + ... = 0 due to f(0) = (0)2 to x0=1 the series are...
  22. M

    MHB How is the remainder for Taylor polynomials calculated?

    Hey! :o I want to calculate the Taylor polynomial of order $n$ for the funktion $ f(x) = \frac{1}{ 1−x}$ for $x_0=0$ and $0 < x < 1$ and the remainder $R_n$. We have that \begin{equation*}f^{(k)}(x)=\frac{k!}{(1-x)^{k+1}}\end{equation*} I have calculated that...
  23. PainterGuy

    How to represent a periodic function using Taylor series

    Hi, Is this possible to represent a periodic function like a triangular wave or square wave using a Taylor series? A triangular wave could be represented as f(x)=|x|=x 0<x<π or f(x)=|x|=-x -π<x<0. I don't see any way of doing although I know that trigonometric series could be used instead...
  24. H Tomasz Grzybowski

    A Transfinite Taylor series of exp(x) and of h(x)

    Let h(h(x)) = exp(x), where h(⋅) is holomorphic in the whole ℂ plane. I want an extension of the domain of exp(⋅) and of h(⋅) so that we can find values of these functions for x = Aleph(0).
  25. Adgorn

    I Understanding the Taylor Expansion of a Translated Function

    I recently found out the rule regarding the Taylor expansion of a translated function: ##f(x+h)=f(x)+f′(x)⋅h+\frac 1 2 h^ 2 \cdot f′′(x)+⋯+\frac 1 {n!}h^n \cdot f^n(x)+...## But why exactly is this the case? The normal Taylor expansion tells us that ##f(x)=f(a)+f'(a)(x-a)+\frac 1...
  26. Velo

    MHB Can't understand/solve Taylor exercises.

    So, the information they give me is the following: $(1) f \in {C}^{3}({\rm I\!R})$ $(2) f(x) = 5 -2(x+2) - (x+2)^2 + (x+2)^3 + R3(x+2)$ $(3) \lim_{{x}\to{-2}} \frac{R3(x+2)}{(x+2)^3}=0$ And they ask me for the equation of the tangent line... Which would be simple if that R3 wasn't there...
  27. Ahmed Abdalla

    Why do we use Taylor expansion expressing potential energy

    My textbook doesn’t go into it, can someone tell me why Taylor expansion is used to express spring potential energy? A lot of the questions I do I think I can just use F=-Kx and relate it to U(x) being F=-Gradiant U(x) but I see most answers using the Taylor expansion instead to get 1/2 kx^2...
  28. D

    Taylor expansion fine structure

    I have to do a Taylor expansion of the energy levels of Dirac's equation with a coulombian potential in orders of (αZ/n)^2 , but the derivatives I get are just too large, I guess there is another approach maybe? This is the expression of the energy levels And i know it has to end like this:
  29. G

    Taylor Series for f(x) = (1+x)^m

    Homework Statement i) What is the Taylor Series for f(x) = (1+x)^m about x=0 where m is a real number? ii) Why does this binomial series terminate when m is a non-negative integer? A iii) Can the result to (i) be used to find the first four non-zero terms of the series for (1+x)^(-1/2)...
  30. A

    Establish Taylor series using Taylor's Theorem in terms of h

    Homework Statement Find the Taylor series for: ln[(x - h2) / (x + h2)] Homework Equations f(x+h) =∑nk=0 f(k)(x) * hk / k! + En + 1 where En + 1 = f(n + 1)(ξ) * hn + 1 / (n + 1)! The Attempt at a Solution ln[(x - h2) / (x + h2)] = ln(x-h2) - ln(x + h2) This is as far as I have been able to...
  31. mertcan

    A Taylor Expansion of Metric Tensor: Troubles & Logic

    Hi, my question is related to taylor expansion of metric tensor, and I have some troubles, I would like to really know that why the RED BOX in my attachment has g_ij (t*x) instead of g_ij(x) ? I really would like to learn the logic...
  32. B

    What is the Angular Frequency of Small Oscillations for a One-Dimensional Mass?

    Homework Statement This is the problem verbatim: The Potential energy of a one-dimensional mass m at distance r from the origin is U(r) = U0 ((r/R) +(lambda^2 (R/r)) for 0 < r < infinity, with U0 , R, and lambda all positive constants. Find the equilibrium position r0. Let x be the...
  33. haushofer

    I Taylor expansions, limits and domains

    Dear all, I have a question concerning calculating the following limit: \lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} \frac{\sin{(x)}}{x} = 1 Obviously, x=0 is not part of the domain of the function. One way to calculate the limit is using l'Hospital. Another way for these kinds of...
  34. S

    A Taylor expansion of dispersion relation - plasma physics

    Hello, I may working through attached paper and really need help with deriving equation in appendix - A4 to give A10. http://iopscience.iop.org/article/10.1088/0004-637X/744/2/182/pdf Any help would be greatly appreciated. thanks, Sinéad
  35. T

    A Questions about Taylor dispersion in laminar pipeflow

    In the equation characterizing the mass transfer in laminar flow, the radial variation of velocity and concentration can be lumped into the axial dispersion term as below: After reading the original paper about Taylor dispersion, I know how to derive this equation. But I am still not able to...
  36. mertcan

    I Why is There a Dot Product in the Taylor Expansion of 1/Distance with Vectors?

    Hi, I would like to express that r and r' are vectors in the attachment and let's say that r is observer distance vector r' is source distance vector. By the way I know this is taylor expansion (for instance if there was only x component (scalar form) I would not any ask question ). But I do...
  37. W

    Justification for upper bound in Taylor polynomial

    Homework Statement I've been reviewing some Taylor polynomial material, and looking over the results and examples here. https://math.dartmouth.edu/archive/m8w10/public_html/m8l02.pdf I'm referring to Example 3 on the page 12 (page numbering at top-left of each page). The question is asking...
  38. K

    B Why Is My Second Derivative Calculation for Taylor Expansion Incorrect?

    So in the book it says expend function ƒ in ε to get following. ƒ=√ (1 + (α + βε)2) = √ (1 + α2) + (αβε)/√ (1 + α2) + (β2ε2)/2 (1 + α2)3/2 + O(e3) When I expend I get(keeping ε = 0) ƒ(0) = √ (1 + α2) -->first term ƒ'(0) = (αβ)/√ (1 + α2) --> sec term with gets multiplied by ε for third...
  39. Batuhan Unal

    Cauchy-Riemann conditions-Multivariable Taylor series

    İ couldn't understand the last operation, please help me.
  40. J

    I Taylor expansions and integration.

    I have a short doubt: Let f(x) be a fuction that can't be integrated in an analytical way . Is anything wrong if I expand it in a Taylor' series around a point and use this expansion to get the value of the definite integral of the function around that point? Suppose that the interval between...
  41. jlmccart03

    What Steps Are Needed to Determine Coefficients in a Taylor Polynomial?

    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > I got my test back and was unable to ask the professor, but how does one solve this problem specifically? I am posting an image of the entire page so you can see my original answers. I just don't...
  42. karush

    MHB 10.8.3 Find the Taylor polynomial

    $\textrm{10.8.{7} Find the Taylor polynomial of orders $0, 1, 2$, and $3$ generated by $f$ at $a$.}$ \begin{align*} \displaystyle f(x)&=\sin{x} \end{align*} \[ \begin{array}{llll}\displaystyle f^0(x)&=\sin{x}&\therefore f^0(\frac{5x}{6})&=\frac{1}{2}\\ \\ f^1(x)&=\cos{x}&\therefore...
  43. K

    Help with Taylor, ln(1-X), |x|<1

    Homework Statement ln(1-X), |x|<1 Homework Equations Could someone verify if it was developed correctly? The Attempt at a Solution ln(1-x) = \sum_{n=0}^\infty \left (a_nx^n\right ) 1+a+a^2+a^3+a^4+a^5+a^6... = 1/(1-a) a=x 1+x+x^2+x^3+x^4+x^5+x^6... = 1/(1-x)...
  44. HOLALO

    B Taylor Polynomials and decreasing terms

    Hi, I have a question about taylor polynomials. https://wikimedia.org/api/rest_v1/media/math/render/svg/09523585d1633ee9c48750c11b60d82c82b315bfI was looking for proof that why every lagrange remainder is decreasing as the order of lagnrange remainder increases. so on wikipedia, it says, for a...
  45. P

    Obtain Taylor Series at x0 = 0?

    Homework Statement Is it possible obtain a Taylor serie at x0=0?Homework Equations f(x)= (\frac{x^4}{x^5+1})^{1/2} [/B] The Attempt at a Solution I think that it is not possible , since f' is not differenciable at x=0, since f' have the factor (\frac{x^4}{x^5+1})^{-1/2} but, for example...
  46. K

    I Taylor series to evaluate fractional-ordered derivatives

    Can the Taylor series be used to evaluate fractional-ordered derivative of any function? I got this from Wikipedia: $$\frac{d^a}{dx^a}x^k=\frac{\Gamma({k+1})}{\Gamma({k-a+1})}x^{k-a}$$ From this, we can compute fractional-ordered derivatives of a function of the form ##cx^k##, where ##c## and...
  47. K

    I Can Taylor series be used to get the roots of a polynomial?

    I'm using this method: First, write the polynomial in this form: $$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x=c$$ Let the LHS of this expression be the function ##f(x)##. I'm going to write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be...
  48. fisher garry

    Usage of Taylor's formula on stress analysis

    Moderator Note: Thread moved from forum Atomic, Solid State, Comp. Physic, so no homework template shown. What function do they use Taylor's formula on? And can you show how they derive it? I know how one derives Taylor formula. Thanks! The text i taken from this...
  49. Archie Medes

    I What methods other than Taylor Series to solve this eq?

    Hi If I have a problem of the form: A1ek1t + A2ek2t = C where A1,A2,k1,k2,C are real and known Or simplified: ex + AeBx = C I can turn it into an nth degree polynomial by Taylor Series expansion, but I'd like to know what other methods I can study Thanks, Archie
  50. F

    Need a book with good intro to Taylor Couette flow

    Hi I just finished a good intro to aerodynamics but I want to understand taylor couette flow. My book stopped at 2d couette flow between 2 plates. I need a good book that someone with only an intro to Aero can understand! Any recommendations?
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