taylor series

  1. m4r35n357

    Python The Wimol-Banlue attractor via the Taylor Series Method

    This attractor is unusual because it uses both the tanh() and abs() functions. A picture can be found here (penultimate image). Here is some dependency-free Python (abridged from the GitHub code, but not flattened!) to generate the data to arbitrary order: #!/usr/bin/env python3 from sys...
  2. Mcp

    B Is there a condition for applying standard limits?

    I came across this basic limits question Ltx->0[(ln(1+X)-sin(X)+X2/2]/[Xtan(X)Sin(X)] The part before '/'(the one separated by ][ is numerator and the one after that is denominator The problem is if I substitute standard limits : (Ltx->0tan(X)/x=1 Ltx->0sin(X)/X=1 Ltx->0ln(1+X)/X=1) The...
  3. m4r35n357

    I How bad is my maths? (numerical ODE method)

    I have written some ODE solvers, using a method which may not be well known to many. This is my attempt to explain my implementation of the method as simply as possible, but I would appreciate review and corrections. At various points the text mentions Taylor Series recurrences, which I only...
  4. S

    Taylor Series Evaluation

    1. Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## 2. Homework Equations The Taylor Series Equation 3. The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except...
  5. Zack K

    Approximating the force on a dipole Taylor series

    1. 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...
  6. Zack K

    I What's the point of Taylor/Maclaurin series?

    We were informally introduced Taylor series in my physics class as a method to give an equation of the electric field at a point far away from a dipole (both dipole and point are aligned on an axis). Basically for the electric field: $$\vec E_{axis}=\frac q {4πε_o}[\frac {1} {(x-\frac s 2)^2}-...
  7. 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...
  8. 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...
  9. A

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

    1. Homework Statement Find the Taylor series for: ln[(x - h2) / (x + h2)] 2. 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)! 3. The Attempt at a Solution ln[(x - h2) / (x + h2)] = ln(x-h2) - ln(x + h2) This is as far as I have...
  10. K

    I Expression of ##sin(A*B)##

    Hi, I've got this: $$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close). Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}## But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...
  11. doktorwho

    Find the limit using taylor series

    1. Homework Statement Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression: $$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$ 2. Homework Equations 3. The Attempt at a Solution ##L=\lim_{x \rightarrow...
  12. Kaura

    Taylor Series Error Integration

    1. Homework Statement Using Taylor series, Find a polynomial p(x) of minimal degree that will approximate F(x) throughout the given interval with an error of magnitude less than 10-4 F(x) = ∫0x sin(t^2)dt 2. Homework Equations Rn = f(n+1)(z)|x-a|(n+1)/(n+1)! 3. The Attempt at a Solution...
  13. T

    I Series Expansion to Function

    I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how. $$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$ I get most of the function, I just can't see where the ##-1## comes from. Could...
  14. S

    I Linearizing vectors using Taylor Series

    I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector. Does that mean I will have to treat it as a Taylor expansion about two variables...
  15. T

    Taylor series representation

    1. Homework Statement Find a power series that represents $$ \frac{x}{(1+4x)^2}$$ 2. Homework Equations $$ \sum c_n (x-a)^n $$ 3. The Attempt at a Solution $$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$ since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} $$...
  16. S

    I Is a function better approximated by a line in some regions?

    I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is...
  17. S

    Taylor Series

    1. Homework Statement Find the Taylor Series for f(x)=1/x about a center of 3. 2. Homework Equations 3. The Attempt at a Solution f'(x)=-x^-2 f''(x)=2x^-3 f'''(x)=-6x^-4 f''''(x)=24x^-5 ... f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n I'm not sure where I went wrong...
  18. Jezza

    Confusing log limit

    1. Homework Statement \lim\limits_{x \to 0} \left(\ln(1+x)\right)^x 2. Homework Equations Maclaurin series: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ... 3. The Attempt at a Solution We're considering vanishingly small x, so just taking the first term...
  19. bananabandana

    I Expansion of ## e^{f(x)} ##

    So, I was doing a question on Laurent series. Part of it asked me to work out the pole of the function: $$ exp \bigg[\frac{1}{z-1}\bigg]$$ The answer is ##1## - since, we can write out a Maclaurin expansion: (1) $$ exp\bigg[\frac{1}{z-1}\bigg] = 1+\frac{1}{z-1}+\frac{1}{2!}\frac{1}{(z-1)^{2}}...
  20. Devin

    Taylor Exansion Series Derivation

    My derivation of Taylor expansion. Hope someone struggling with it gets use!
  21. T

    Expanding a function in terms of a vector

    1. 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...
  22. J

    Proof Taylor series of (1-x)^(-1/2) converges to function

    Hello, I want to prove that the taylor expansion of f(x)={\frac{1}{\sqrt{1-x}}} converges to ƒ for -1<x<1. If I didn't make a mistake the maclaurin series should look like this: Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n My attempt is to use the lagrange error bound, which is...
  23. A

    Conceptual: Are all MacLaurin Series = to their Power Series?

    1. Homework Statement To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do...
  24. END

    How do we define "linear" for single and multivariable?

    In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following: $$x^2, y^2, xy$$ What surprised me was the fact that xy was listed as a nonlinear term. In...
  25. C

    Statistical Mechanics Mean Field Model

    I don't think I've fully grasped the underlying ideas of this class, so at the moment I'm just sort of flailing for equations to plug stuff into... 1. Homework Statement Show that in the mean field model, M is proportional to H1/3 at T=Tc and that at H=0, M is proportional to (Tc - T)1/2 2...
  26. T

    Estimate number of terms needed for taylor polynomial

    1. Homework Statement For ln(.8) estimate the number of terms needed in a Taylor polynomial to guarantee an accuracy of 10^-10 using the Taylor inequality theorem. 2. Homework Equations |Rn(x)|<[M(|x-a|)^n+1]/(n+1)! for |x-a|<d. 3. The Attempt at a Solution All I've done so far is take a...
  27. A

    Taylor's Theorem

    I am studying power series right now and I am understanding well how to write them and where they converge but I am having some trouble grasping the Taylor Remainder Theorem for a few reasons. First of all it says the remainder is: f^(n+1)(c)(x-a)^(n+1)/(n+1)! for some c between a and x. I...
  28. A

    Taylor Series Question

    I am just trying to clarify this point which I am unsure about: If I am asked to write out (for example) a third order taylor polynomial for sin(x), does that mean I would write out 3 terms of the series OR to the x^3 term. x-x^3/3!+x^5/5! or just x-x^3/3! Also, I have a question for the...
  29. T

    Taylor Polynomial of 3rd order in 0 to f(x) = sin(arctan (x))

    The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand. Using the Taylor series we will write sin(x) as: sin(x) = x - (x^3)/6 + (x^5)B(x) and...
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