What is Taylor series: Definition and 492 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. O

    Taylor Series of (\pi - x)^-2 around a = 0

    Homework Statement Write the Taylor series of the function f(x) = (\pi -x)^-2 around a = 0 Homework Equations (\pi - x)^-2 = f(a) + f'(a)(x-a) + [f''(a)(x-a)^2]/(2!) +...+ [f^n(a)(x-a)^n]/(n!) The Attempt at a Solution This is what i have and i am not sure i am showing it...
  2. S

    Improving Cosine Approximation Using Taylor Series in Matlab

    Homework Statement Write a user-defined function that determines cos(x) using Taylor Series expansion Stop adding terms when estimated error, E<=.000001 Homework Equations sum Sn = Sn-1 + an E = | (Sn - Sn-1)/Sn-1 | The Attempt at a Solution function y = cosTaylor(x) Sn=1...
  3. P

    Numerical Methods: Taylor Series for Diff Equation

    Homework Statement Solve the differential equation \frac{dy^2}{dx^2}=xy^2-2yy'+x^3+4 where y(1)=1 y'(1)=2 by means of the Taylor-series expansion to get the value of y at x=1.1. Use terms up to x^6 and \Delta x=0.1The Attempt at a Solution I'm unsure as to how I should go about...
  4. H

    Taylor Series Expansion for f(t) and G(x) with Convergence Analysis"

    Let f be the function given by f(t) = 4/ (1 + t^2) and G be the function given by G(x) = {Integral from 0 to x} f(t)dt . (a) Find the first four nonzero terms and the general term for the power series expansion of f(t) about t = 0. (b) Find the first four nonzero terms and the general term...
  5. M

    How Do You Calculate the nth Coefficient of a Taylor Series for sqrt(x) at a=1?

    Homework Statement I am trying to find the Tn(x) for sqrt[x] centered at a=1 Homework Equations The Attempt at a Solution right now i have f'(x)=1/2x^-1/2 f''(x)=-1/4x^-3/2 f'''(x)=3/8x^-5/2 f''''(x)=-15/16x^-7/2 f'(1)=1/2 f''(1)=-1/4 f'''(1)=3/8 f''''(1)=-15/16 how...
  6. S

    Rewriting Taylor Series in Sigma Notation

    Homework Statement I understand the whole concept of Taylor Series and Maclaurin series but I don't know how to rewrite them in sigma notation. I'll use this generic example. Find the Maclaurin series of the function \ f(x)=e^{x} Homework Equations The Attempt at a Solution \...
  7. S

    How Do You Correctly Apply Taylor Series Expansion for f(x-dx)?

    Hi, how would you find the taylor series for f(x-dx). i know that substituting x-dx in the series for f(x) is not correct.
  8. T

    MATLAB MATLAB Help for expansion of cos(x) using a Taylor Series

    I was hoping somebody would be able to help me as I am pretty new to Matlab. I am trying to create a for-loop to describe the taylor series expansion of cos(x)= (-1)^n*x^2n/(2n)! and to see how it converges towards cos(x). Below is the code that I have used to plot the different orders of n, but...
  9. L

    Taylor Series Expansion for the Relativistic Factor of Momentum

    Homework Statement Using the technique of Taylor expansion, find an approximate expression for the relativistic factor γ for small v (i.e., expanded around v = 0) that is correct to order v2. Homework Equations γ=1/SQRT(1+ V2/C2). But in class, my professor just substituted X=V/C, so...
  10. N

    Why Do Taylor Series Representations of Cosine Use Alternating Powers of -1?

    when i develop the series of a cosine i have a (-1) member i wanted to represent the series as a sum so i need to take only the odd members so the power of -1 is 2k+1 i got but the solution says that the power of -1 is equal (-1)^{k-1} is it the same?? why they have such an expression...
  11. J

    How Can I Find the Equation for a Functional Taylor Expansion?

    Hello, Is there any place I can find the equation for the Taylor expansion of a functional around a function ?? Particularly, I want something like: f[x(t)] = f[\hat{x}(t)] + (f[\hat{x}(t)] - f[x(t)] \frac{\delta f}{\delta x(t)}|_{x(t)=\hat{x}(t)} + \frac{(f[\hat{x}(t)] -...
  12. I

    Understanding the Remainder Term in Taylor Series: A Closer Look at the Formula

    I'm currently studying the Taylor series and I cannot figure out how the remainder term came to be. If anyone could clarify this for me, I would be really grateful ...! I understand that the Taylor series isn't always equal to f(x) for each x, so we put Rn at the end as the remainder term...
  13. S

    MATLAB Taylor Series without using the built-in MATLAB Taylor's Function

    [URGENT] Taylor Series without using the built-in MATLAB "Taylor's Function" I have a MATLAB Test Tomorrow Please teach me the MATLAB programming to solve Taylor & Maclaurin Series, without using the built-in MATLAB "Taylor's Function" Please explain the procedure to solve them using the...
  14. C

    What is the Problem with the Taylor Series for f(x)=1/(x)^(1/2) at a=9?

    Homework Statement Find the taylor series of f(x)=1/(x)^(1/2) ; a=9 2. The attempt at a solution f(x) = (x)^(-1/2) f'(x) = -(1/2)*x^(-3/2) f''(x) = (1/2)*(3/2)*x^(-5/2) f'''(x) = -(1/2)*(3/2)*(5/2)*x^(-7/2) f''''(x) = (1/2)*(3/2)*(5/2)*(7/2)*x^(-11/2) f(9) =...
  15. B

    Discovering the Taylor Series for cos(x) at PI: Finding the Right Pattern

    Trying to find the Taylor Series for cos(x) where x0 is PI. I've gotten cos(x) -1 -sin(x) 0 -cos(x) 1 sin(x) 0 cos(x) -1 It's clearly 0 every other term so I need 2k or 2k-1. But the -1 term switches between -1 and 1 How in world do I deal with this? xD Thanks for any...
  16. V

    Calculating the Taylor Series for Arctan(x): Explained and Illustrated

    The series is: (33/5) - (34/7) + (35/9) - (36/11)+... Looking at this, I'm guessing I can use the Taylor Series for arctan(x) but I don't know how to apply it or where to begin. Any help is greatly appreciated.
  17. V

    Taylor Series Help: Solving sin(x) Equation

    The Taylor Series of sin(x)=x-(x3/3!)+(x5/5!)-... What function of sin gives the following: (\pi2/(22) - (\pi4/(24*3!)+ (\pi6/(26*5!) - (\pi8/(28*7!)+... Please help me. Thank you.
  18. V

    Can the Taylor Series Method Accurately Compute Integrals with 10-3 Precision?

    Use taylor series method to compute the integral from 1 to 2 of [sin(x2)] / (x2) with 10-3 precision.
  19. V

    Solving Taylor Series: Discover the Function Behind this Tricky Sequence

    Homework Statement What function produces the following: (\pi2/(22)) - (\pi4/(24*3!)) + (\pi6/(26*5!)) - (\pi8/(28*7!)) I'm sure this is a sin function. But I can't figure out what exactly is the function. Please help.
  20. V

    How Can Taylor Series Be Used to Compute Integrals with High Precision?

    Homework Statement Use taylor series method to compute the integral from 1 to 2 of [sin(x2)] / (x2) with 10 -3 precision Homework Equations The Attempt at a Solution I'm not sure where to start. Someone please help me.
  21. C

    How Do You Find a Taylor Series for the Square Root of X About c=1?

    Homework Statement Find a taylor series for f(x)=sq. rt. of X about c=1 Homework Equations N/A The Attempt at a Solution I took the derivative of the sq rt of X, and then plugged in 1 for all the X's. I got: f(x)= 1 f'(x)=1/2 f''(x)=-1/4 f'''(x)=3/8 f^4(x)=-15/16 My teacher...
  22. U

    When a Taylor Series Converges

    Homework Statement For what values of x do you expect the following Taylor series to converge? sqrt(x^{2}-x-2) Homework Equations I'm not too sure The Attempt at a Solution Well quite frankly I have no idea what to do. If someone can push me in the right direction I'll get the rest done.
  23. B

    Convergence of Taylor Series for Various Functions

    Homework Statement For what values of x (or \theta or u as appropriate) do you expect the following Taylor Series to converge? DO NOT work out the series. \sqrt{x^{2}-x-2} about x = 1/3 sin(1-\theta^{2}) about \theta = 0 tanh (u) about u =1 Homework Equations The...
  24. H

    Optimal Degree for Approximating Cosine with Taylor Series

    Homework Statement What degree Taylor Polynomial around a = 0(MacLaurin) is needed to approximate cos(0.25) to 5 decimals of accuracy? Homework Equations taylor series...to complicated to type out here remainder of nth degree taylor polynomial = |R(x)| <= M/(n+1)! * |x - a|^(n+1)...
  25. J

    Taylor series vs. Fourier series

    Is a Fourier series essentially the analogue to a Taylor series except expressing a function as trigs functions rather than as polynomials? Like the Taylor series, is it ok only for analytic functions, i.e. the remainder term goes to zero as n->infinity?
  26. J

    Taylor series for differential equation solution

    Homework Statement Find the series solution for: y'=x^2-y^2,y(1)=1 Homework Equations The Attempt at a Solution I have correctly derived the series solution as: y(x)=1+(x-1)^2-\frac{(x-1)^3}{3}+\frac{(x-1)^4}{6}-... But I cannot get the book solution for the INTERVAL OF...
  27. I

    Function can be represented by a Taylor series

    If a function can be represented by a Taylor series at x0, but only at this point, (radius of convergence = 0), is it considered analytic there?
  28. I

    Calculating Taylor Series for $\frac{1}{|R-r|}$ with R>>r

    I can't work out how to calculate the Taylor series for \frac{1}{|R-r|} when R>>r, but they are both vectors. We were told to expand in r/R but I did the step below and I'm not sure where to go from there I got to \frac{1}{R \sqrt{1 - (2R.r)/R^2 + (r^2)/(R^2)}} I also know the result...
  29. S

    Berthelot equation of state - virial coefficient and taylor series

    Using the taylor series result Vm / Vm - b = 1 + b / Vm + ... and the definition of hte compressibility factor Z = PVm / RT, derive an expression for the first virial coefficient in terms of a and b for the Berthelot equation of state.
  30. C

    Help with Taylor Series Project

    Hi, My lecture had gave a project about analyzing and discussion about - Taylor Series. I had done some research and tried understand and solve the question, but I'm in trouble now. I could only complete No.1 and No.2 (don't know whether is correct or not), I stuck at No.3 I have no idea...
  31. J

    Deriving Planck's law with Taylor series

    Expanding exp(hc / lambda*k_b * T) by Taylor series = 1 + hc /lambda*k_B * T +... But don't you take the derivative with respect to lambda? So I don't get how it would be this.
  32. N

    Deriving a Taylor Series for Sinx: Is it the Same as a Power Series?

    Is it correct to take the derivative of a taylor series the same as you would for a power series ie: sinx=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} \frac{d}{dx}(sinx)=cosx=\sum_{n=1}^{\infty}(-1)^n(2n+1)\frac{x^{2n}}{(2n+1)!} it seems as if it wouldn't be...
  33. J

    Does a Taylor Series Exist for f(x)=tanh(x)/x and f(x)=ln(1+x)/x?

    Is it correct that a taylor series does not exist for f(x)=tanh(x)/x and f(x)=ln(1+x)/x. I differentiated to f'''(x) and fn(0) and all equal zero.
  34. J

    Find Taylor Series of \frac{1/3}{1-2x^3/3}

    Homework Statement Find the Taylor series about the point x = 0 for the function \frac{1}{3-2x^3} Homework Equations The Attempt at a Solution \frac{1}{3 - 2x^3} = \frac{1}{3(1 - \frac{2x^3}{3})} . Let u = \frac{2x^3}{3} . Then \frac{1}{3(1 - \frac{2x^3}{3})} = \frac{1}{3} \frac{1}{1 - u} =...
  35. G

    Linear approximations derived from Taylor series

    Homework Statement So I have the problem questiona dn my teachers solution posted below. I understand: f(xo) = sin pi/6 f '(xo) = cos pi/6 but i don't know how he gets them into fraction form with the SQRT of 3, it looks like some pythagoras but i don't really know how he did it...
  36. M

    Exploring Taylor Series for f(x) and g(x)

    ive got a question to ask I am working on taylor series and want to know f(x)=In(3+x) and g(x)=In (1+x) by writing In(3+x)=In3+In(1+1/3x) im asked to use substitution in one off the standard taylor series given in the course.to find about 0 for f explicitly all...
  37. R

    Can Factoring Out a Negative One Affect the Convergence of a Taylor Series?

    First of all if i have a function with all negative terms is it possible to determine its convergence simply by factoring the negative one, treating the other terms as a positive series determine its convergence then assume that multiplying by the constant negative one will not change its...
  38. B

    Solving Goldstein 3.3: Taylor Series & Newton-Rhapson

    Homework Statement (Goldstein 3.3) If the difference \psi - \omega t in represented by \rho, Kepler's equation can be written: \rho = e Sin(\omega t + \rho) Successive approximations to \rho can be obtained by expanding Sin(\rho) in a Taylor series in \rho, and then replacing \rho...
  39. R

    Find the function for this Taylor series

    Find the function that has the following Taylor series representation: \sum^{\infty}_________{m=0}\frac{(m+s)^{-1}x^{m}}{m!} Where s is a constant such that 0<Re(s)<1. Any ideas?
  40. Mapes

    Laplace transform of a Taylor series expansion

    I'm reading a paper on tissue cell rheology ("Viscoelasticity of the human red blood cell") that models the creep compliance of the cell (in the s-domain) as J(s) = \frac{1}{As+Bs^{a+1}} where 0\leq a\leq 1. Since there's no closed-form inverse Laplace transform for this expression, they...
  41. A

    Taylor series radius of convergence and center

    When approximating a function with a Taylor series, I understand a series is centered around a given point a, and converges within a certain radius R. Say for a series with center a the interval of convergence is [a-R, a+R]. Does this imply that: 1. There also exists a Taylor series expansion...
  42. E

    Book containing taylor series expansions

    Hello, I am looking for a resource (preferably a textbook) to help me with nonlinear, multivariable functions and working through taylor series expansions of them. My calculus book only covers single variable expansions unfortunately. Thanks
  43. A

    Question about Taylor series and big Oh notation

    Question about Taylor series and "big Oh" notation Can someone please explain WHY it's true that e^x = 1 + x + \frac{x^2}{2} + \mathcal{O}(x^3) I'm somewhat familiar with "big Oh" notation and what it stands for, but I'm not quite sure why the above statement is true (or statements...
  44. M

    Taylor series with summation notation

    Homework Statement f(x) = \frac{1-cos(X^2)}{x^3} which identity shoud i use? and tips on this type of questions? once i can separate them, then i'll be good thanks!
  45. L

    Taylor Series Expansion of Analytic Function at x0 = 0

    you know this, right? f(x) = \sum^{\infty}_{k=0} \frac{f^{(k)}(x_0) (x-x_0)^k}{k!} for an analytic function, at x0 = 0, you have to say that 0^0 equals 1 for the constant term. if 0^0 is indeterminate then how can you just say it's 1 in this case?
  46. R

    Taylor series of two variable ?

    Homework Statement I want to know that how to calculate the required number of terms to obtain a given decimal accuracy in two variable Taylor series . In one variable case i know there is an error term R(n)=[ f(e)^(n+1)* (x-c)^(n+1)] / (n+1)! where 'e' is...
  47. C

    Expanding A Taylor series.

    [b]1. Hi, I am new to taylor series expansions and just wondered if somebody could demonstrate how to do the following. Find the Taylor series of the following functions by using the standard Taylor series also find the Radius of convergence in each case. 1.log(x) about x=2...
  48. K

    How Does Taylor's Theorem Apply to Logarithmic Series?

    Homework Statement (a) Use Taylor's theorem with the Lagrange remainder to show that log(1+x) = \sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}x^{k} for 0<x<1. (b) Now apply Taylor's theorem to log(1-x) to show that the above result holds for -1<x<0. Homework Equations Taylor's...
  49. J

    Finding Taylor series about some point

    In this: http://www.math.tamu.edu/~fulling/coalweb/sinsubst.pdf It says that to find the Taylor series of sin(2x + 1) around the point x = 0, we cannot just substitute 2x+1 into the Maclaurin series for sinx because 2x + 1 doesn't approach a limit of 0 as x approaches 0. It says we have...
  50. J

    Estimate Remainder of Taylor Series

    1. The problem \statement, all variables and given/known data Estimate the error involved in using the first n terms for the function F(x) = \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I am using the Lagrange form of the remainder. I need to know the n+1 derivative of...
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