What is Taylor: Definition and 873 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. I

    Finding the Coefficients of a Taylor Polynomial: A Tricky Integration Question

    Homework Statement Here's a screenshot of the problem: http://puu.sh/2Bta5 Homework Equations The Attempt at a Solution As can be seen by the screenshot, the answer's already given, but I'm not sure how to go about getting it. This one has me stumped since e-4x and sin(5x) are...
  2. L

    Compute the second order Taylor polynomial centered at 2 for ln(x)

    Homework Statement a. Compute the second order Taylor polynomial centered at 2, P2(x), for the function ln(x). b. Estimate the maximum error of the answer to part a for x in the interval [1,2].Homework EquationsThe Attempt at a Solution For part a, I'm thinking that when it says "second...
  3. L

    Find Taylor series generated by e^x centered at 0.

    1. a. Find Taylor series generated by ex2 centered at 0. b. Express ∫ex2dx as a Taylor series. 2. For part a, I just put the value of "x2" in place of x in the general form for the e^x Taylor series: ex: 1 + x + x2/2! + x3/3! + ... ex2: 1 + x2 + x4/2! + x6/3! + ... For part b...
  4. S

    Why is the correction important?

    Homework Statement Find the Taylor series of f(x) = x2ln(1+2x2) centered at c = 0. Homework Equations Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)n-1xn/n The Attempt at a Solution I have worked the problem to (-1)n4nx2n/n I am not sure where to go from here...
  5. S

    Taylor Series of f(x) = 1/(1-6x) at c=6

    Homework Statement Find the Taylor Series for f(x) = 1/(1-6) centered at c=6 Homework Equations ∞ Ʃ Fn(a)(x-a)/n! n=0 The Attempt at a Solution I believe that the nth derivative of 1/(1-6x) is (-6)n-1n!/(1-6x)n+1 So i figured that the taylor series at c=6 would be...
  6. X

    Understanding Taylor Series for Solving Complex Equations

    Homework Statement f(x)=\frac{4x}{(4+x^{2})^{2}}Homework Equations \frac{1}{1-x} = \sum x^{n} The Attempt at a Solution How am I supposed to use that equation to solve the main problem. I have the solution but I don't understand how to do any of it. My professor is horrible, been on...
  7. 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...
  8. S

    Understanding the Taylor Series of e^x/(x-1)

    Homework Statement Let g(x) = \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} \frac{B_n}{n!} x^n be the taylor series for g about 0. Show B_0 = 1 and \sum_{k=0}^{n} \binom{n+1}{k} B_k = 0 .Homework Equations The Attempt at a Solution g(x) = \sum_{n=0}^{\infty} \frac{g^{(n)}(0)}{n!} x^n , but...
  9. STEMucator

    Integrating Taylor Series for Sine Functions

    Homework Statement A problem from advanced calculus by Taylor : http://gyazo.com/5d52ea79420c8998a668fab0010857cf Homework Equations ##sin(x) = \sum_{n=0}^{∞} (-1)^n \frac{x^{2n+1}}{(2n+1)!}## ##sin(3x) = \sum_{n=0}^{∞} (-1)^n \frac{3^{2n+1}x^{2n+1}}{(2n+1)!}## The Attempt at a Solution...
  10. 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...
  11. STEMucator

    Calculating Taylor Series Expansion for ##f(x)##

    Homework Statement Calculate the Taylor series expansion about x=0 as far as the term in ##x^2## for the function : ##f(x) = \frac{x-sinx}{e^{-x} - 1 + ln(x+1)}## when ##x≠0## ##f(x) = 1## when ##x=0## Homework Equations Some common Taylor expansions. The Attempt at a Solution...
  12. T

    Is there any benefit to using Taylor series centered at nonzero value

    over a Maclaurin series? Also, how do I calculate e^0 using Maclaurin series? I'm getting 0^0.
  13. 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) =...
  14. S

    Why Taylor Series works so well for some functions and not for others

    About a week ago, I learned about linear approximation from a great youtube video, it was by Adrian Banner and the series of his lectures I think were from his book Calculus LifeSaver. I truly thought it was so beautiful and powerful a concept. Shortly I also got to know the Taylor Series and...
  15. S

    Taylor polynomial remainder term

    Homework Statement Consider the followign function f(x) = x^-5 a=1 n=2 0.8 \leq x \leq 1.2 a) Approximate f with a tayloy polynomial of nth degree at the number a = 1 b) use taylor's inequality to estimate the accuracy of approximation f(x) ≈ T_{n}(x) when x lies in the interval...
  16. A

    Maximizing Planck's law using Taylor polynomial for e^x

    Homework Statement The energy density of electromagnetic radiation at wavelength λ from a black body at temperature T (degrees Kelvin) is given by Planck's law of black body radiation: f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)} where h is Planck's constant, c is the speed of light, and...
  17. K

    Messy Taylor polynomial question

    Homework Statement Find the Taylor polynomial approximation about the point ε = 1/2 for the following function: (x^1/2)(e^-x)The Attempt at a Solution I'm trying to get a taylor polynomial up to the second derivate i.e.: P2(×) = (×^1/2)(e^-x) + (x-ε) * [(e^-x)(1-2×)/2(×^1/2)] +...
  18. E

    Taylor series for getting different formulas

    I am trying to establish why, I'm assuming one uses taylor series, \frac{\partial u}{\partial t}(t+k/2, x)= (u(t+k,x)-u(t,x))/k + O(k^2) I have tried every possible combination of adding/subtracting taylor series, but either I can not get it exactly or my O(k^2) term doesn't work out (it's...
  19. I

    MHB Proving $f(x_k+εp)<f(x_k)$ with Taylor Series

    Prove that if $p^T▽f(x_k)<0$, then $f(x_k+εp)<f(x_k)$ for $ε>0$ sufficiently small. I think we can expand $f(x_k+εp)$ in a Taylor series about the point $x_k$ and look at $f(x_k+εp)-f(x_k)$, but what's then? (Taylor series: $f(x_0+p)=f(x_0)+p^T▽f(x_0)+(1/2)p^T▽^2f(x_0)p+...$ => here...
  20. F

    Taylor series expansion of Dirac delta

    I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...
  21. I

    Finding a Taylor Series from a function and approximation of sums

    Homework Statement \mu = \frac{mM}{m+M} a. Show that \mu = m b. Express \mu as m times a series in \frac{m}{M} Homework Equations \mu = \frac{mM}{m+M} The Attempt at a Solution I am having trouble seeing how to turn this into a series. How can I look at the given function...
  22. S

    Quick Question on Taylor Expansions

    Hello all, I am a senior physics undergraduate student. I have wondered about the Taylor Expansion for a few years now and just have never bothered to ask. But I will now: I know the Taylor Expansion goes like: f(a) + \frac{f'(a)}{1!}*(x-a) + \frac{f''(a)}{2!}*(x-a)^{2} +...
  23. Z

    Taylor Polynomial approximation

    Homework Statement obtain the number r = √15 -3 as an approximation to the nonzero root of the equation x^2 = sinx by using the cubic Taylor polynomial approximation to sinxHomework Equations cubic taylor polynomial of sinx = x- x^3/3!The Attempt at a Solution Sinx = x-x^3/3! + E(x) x^2 =...
  24. W

    Understanding the Error of Taylor Polynomials in Approximating Functions

    the error of a taylor series of order(I think that's the right word) n is given by \frac{f^{n+1} (s)}{n!} (x-a)^n I think this is right. The error in a linear approximation would simply be \frac{f''(s)}{2} (x-a)^2 My question is what is s and how do I find it. Use linear...
  25. 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...
  26. micromass

    Classical What is the best undergraduate book on Classical Mechanics?

    Author: John Taylor Title: Classical Mechanics Amazon Link: https://www.amazon.com/dp/189138922X/?tag=pfamazon01-20 Prerequisities: A Lower-Division mechanics course Contents: Upper-Division of undergrad
  27. Greg Bernhardt

    Relativity Spacetime Physics by Edwin F. Taylor and John Archibald Wheeler

    Author: Edwin F. Taylor (Author), John Archibald Wheeler (Author) Title: Spacetime Physics Amazon Link: https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20 Prerequisities: Contents:
  28. J

    Taylor Series Problem - Question and my attempt so far

    Question: http://i.imgur.com/GsjeL.png Here is my attempt so far: http://i.imgur.com/AyOCm.png Note: I've used m where the question has used j. My attempt is based off some bad notes I took in class so the way I am trying to solve the problem may not be the best. I'm struggling to...
  29. 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.
  30. C

    Taylor series question

    The Taylor Series of f(x) = exp(-x^2) at x = 0 is 1-x^2... Why is this? The formula for Taylor Series is f(x) = f(0) + (x/1!)(f'(0)) + (x^2/2!)(f''(0)) + ... and f'(x) = -2x(exp(-x^2)) therefore f'(0) = 0? Can someone please explain why it is 1-x^2?
  31. C

    Understanding and Solving the Taylor Series for a Specific Point

    What does it mean to calculate the Taylor series ABOUT a particular point? I understand the formula for the Taylor series but how do you solve it about a particular point for a function? It's the about the particular point that confuses me. Could someone please explain this and provide...
  32. 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)...
  33. M

    Why the Taylor Series has a Factorial Factor

    Why in Taylor series we have some factoriel ##!## factor. f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+... Why we have that ##\frac{1}{n!}## factor?
  34. T

    On Taylor Series Expansion and Complex Integrals

    I'm trying to understand how to use Taylor series expansion as a method to solve complex integrals. I would appreciate someone looking over my thoughts on this. I don't know if they are right or wrong or how they could be improved. I suppose that my issue is that I don't feel confident in my...
  35. F

    Compute Tricky Limit Using Taylor Series and De L'hopital's Theorem

    Homework Statement compute the following limit: ## \displaystyle{\lim_{x\to +\infty} x \left((1+\frac{1}{x})^{x} - e \right)} ## The Attempt at a Solution i wanted to use the taylor expansion, but didn't know what ##x_0## would be correct, as the x goes to ## \infty##. also, i tried to...
  36. P

    How to find the cosine of i using Taylor series?

    Is there a way to find the cosine of i, the imaginary unit, by computing the following infinite sum? cos(i)=\sum_{n=0}^\infty \frac{(-1)^ni^{2n}}{(2n)!} Since the value of ##i^{2n}## alternates between -1 and 1 for every ##n\in\mathbb{N}##, it can be rewritten as ##(-1)^n##...
  37. B

    Taylor Polynomial. Understanding.

    Homework Statement Last exam in my school this exircise was given: From norweagen: " Decide the Taylor polynomial of second degree of x=0 of the function: f(x) = 3x^3 + 2x^2 + x + 1 I found the Taylor polynomial of second degree to be: 2X^2+X+1, which is correct. If I get an...
  38. A

    Taylor Approximation: Error Calculation Tool?

    Often you use taylor series to approximate differential equations for easier solving. An example is the small angle approximation to the pendulum. My question is: Is there mathematical tool for calculating the error you make as time goes with such an approximation? Because I could Imagine it...
  39. H

    How to write taylor series in sigma notation

    Homework Statement My Calc II final is tomorrow, and although we never learned it, it's on the review. So I have a few examples. Some I can figure out, some I cant. Examples: f(x)=sinh(x), f(x)=ln(x+1) with x0=0, f(x)=sin(x) with x0=0, f(x)=1/(x-1) with x0=4 The only one of those that I was...
  40. 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...
  41. R

    Taylor Polynomial of Smallest Degree to approximation

    Hey, guys. Having problems with this question because I don't exactly know how to begin it. Homework Statement The problem states to: "Find the Taylor polynomial of smallest degree of an appropriate function about a suitable point to approximate the given number to within the indicated...
  42. P

    Modifying taylor series of e^x

    I recently thought to myself about how a slight modification to the taylor series of e^x, which is, of course: \sum_{n=0}^\infty \frac{x^n}{n!} would change the equation. How would changing this to: \sum_{n=0}^\infty \frac{x^{n/2}}{\Gamma(n/2+1)} change the equation? Would it still be...
  43. R

    Polylogarithm and taylor series

    let nε Z. the polylogarithm functions are a family of functions, one for each n. they are defined by the following taylor series: Lin(x)= Ʃ xk/kn 1.calculate the radius of convergence [b]3. when i attempted this part, i couldn't use theratio or root test, so by comparison i got R=∞...
  44. O

    MHB Taylor Series: Exploring Properties & Applications

    Hello Everyone! Suppose $f(x)$ can be written as $f(x)=P_n(x)+R_n(x)$ where the first term on the RHS is the Taylor polynomial and the second term is the remainder. If the sum $\sum _{n=0} ^{\infty} = c_n x^n$ converges for $|x|<R$, does this mean I can freely write $f(x)=\sum _{n=0} ^{\infty}...
  45. 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...
  46. E

    Approximation sin(x) taylor Series and Accuracy

    Homework Statement One uses the approximation sin(x) = x to calculate the oscillation period of a simple gravity pendulum. Which is the maximal angle of deflection (in degree) such that this approximation is accurate to a) 10%, b) 1%, c) 0.1%. You can estimate the accuracy by using the next...
  47. P

    MHB Finding $a_n$s for $f(z)$ in a Taylor Series

    Consider the function $$f(z)=e^{\frac{1}{1-z}}$$ It has an essential singularity at $z_0=1$ and hence it can be expanded in a Laurent series at $z_0$. But I'm interested in Taylor expansion. The function is analytic in the unit open disc at the origin, so I'm looking for $a_n$ where...
  48. P

    Aproximating a morse potential using a taylor polynomial

    I am not going to post my question because I want to find out how to actually use the taylor polynomial and morse potential and then apply that to my problem. Say I have to approximate the morse potential using a taylor series expanding about some value. This will then find me the force...
  49. A

    A question about Taylor Series

    Find the Taylor series for cosx and indicate why it converges to cosx for all x in R. The Taylor series for cosx can be found by differentiating sum_{k=0}^{\infty} \frac{(-1)^k (x^{2k+1})}{(2k+1)!} on both sides... But I'm not sure what the question means by "why it converges to cosx for...
  50. M

    Approximating accuracy of Taylor polynomials

    Homework Statement Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. e^x ≈ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} For x < 0 Homework Equations Taylor's Theorem to approximate a remainder: |R(x)| =...
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