1. Limited time only! Sign up for a free 30min personal 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 Series and Maclaurin Series Doubt

  1. Oct 9, 2012 #1
    1. The problem statement, all variables and given/known data
    If I take a function f(x) and its taylor series, then will the infinite series give me the value of the function at any x value or will it only give proper values for x≈a?

    For example, If I take a maclaurin series for a function will it give me proper values for all x values or only if x≈a=0?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 9, 2012 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It always works for x = a. It is possible that is the only value for which it works. But more usually there is a radius of convergence where it works for |x-a|<r and the series diverges for |x-a|>r. Or it may converge and equal the function for all x. Generally, the farther you are from x = a, the more terms you need for a given accuracy. There are examples where the series converges for all x but doesn't equal the function except at x = a.
     
  4. Oct 9, 2012 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    There exist functions for which the Taylor series exist and converges for all x but converges to the value of the function only at the given point. The function defined as [itex]f(x)= e^{-1/x^2}[/itex] for x not equal to 0, 0 for x= 0, has derivatives of all order and they are all 0 at x= 0 so the Maclaurin series (Taylor series with a= 0) is just 0 for all x. But f is 0 only at x= 0.

    (Functions that have the property that they are equal to their Taylor series at every point on a set are called 'anlytic' on that set. Those are especially important in functions of complex numbers.)
     
  5. Oct 10, 2012 #4
    So how could we find out whether the sum will give me the approximate value of the function or not???
     
  6. Oct 10, 2012 #5

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Take a look at:

    http://en.wikipedia.org/wiki/Taylor's_theorem
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Taylor Series and Maclaurin Series Doubt
Loading...