1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Taylor's theorem: What does the a represent?

  1. Sep 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Taylor's theorem states:
    [tex]f(x)=\displaystyle\sum_{k=0}^{infinity}{\frac{(f(^k(a)(x−a)^k)}{n!}}[/tex]

    When approximating a function, we always set a=0, and it gives an approximation for the whole function.

    Why set the a=0? What does the a reperesent?
    Thank you.

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Sep 22, 2012
  2. jcsd
  3. Sep 22, 2012 #2
    You don't always set [itex]a=0[/itex]. You can set [itex]a[/itex] to whatever number you want. In all cases, the point [itex]a[/itex] is the point that you expand the series about--a point that, in principle, you should know several derivatives of the function at in order to construct an approximation. There is no reason this point must be the origin, though often times, it is.

    For example, in physics one often Taylor expands around the bottom of a potential well. Even if the well is not centered on the origin, this is a valid thing to do. One takes [itex]a[/itex] to be where the minimum of the well lies, and immediately, one knows that [itex]f'(a) = 0[/itex] because the point is a minimum. This derivative being zero is equivalent to saying the force at an equilibrium point is zero, and knowing that the first non-vanishing term in the expansion is quadratic in [itex]x-a[/itex] tells us that, very close to the minimum, any potential can be approximated as a simple harmonic potential, which also goes as [itex](x-a)^2[/itex].
     
  4. Sep 22, 2012 #3

    chiro

    User Avatar
    Science Advisor

  5. Sep 22, 2012 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Your statement "When approximating a function, we always set a=0, and it gives an approximation for the whole function" is false: we do not always set a = 0, and even when we do we do not necessarily approximate the "whole function". For example,
    [tex] \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots, [/tex] but this is valid only if -1 < x < 1. So, if you want to expand f(x) = 1/(1-x) in a power series near x = 2, for example, you would need to expand in powers of (x-2), not x. In fact,
    [tex] \frac{1}{1-x} = \frac{1}{-1 + 2-x} = \frac{-1}{1 + (x-2)} = -1 + (x-2) - (x-2)^2 + \cdots.[/tex] This is valid for |x-2| < 1, or 1 < x < 3.

    RGV
     
    Last edited: Sep 22, 2012
  6. Sep 22, 2012 #5
    Okay, thank you! so basically, it is the estimation around the point x=a, and the further it deviates from a the more inaccurate? For example, evaluating the value of e, without using a calculator, we still set a=0, because then[tex]f^n(a)[/tex] will be 1, and it will be easy to calculate, though there will be an error term?
     
  7. Sep 22, 2012 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The infinite series has error = 0. However, if you have a non-polynomial function you will get an error if you do not actually include all of the infinitely many terms.

    RGV
     
  8. Sep 22, 2012 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    By the way, the special case of Taylor's series using the specific value a= 0, is called the "MacLaurin series".
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Taylor's theorem: What does the a represent?
  1. Taylor's Theorem (Replies: 1)

Loading...