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Math Help - Maclaurin/Taylor/Power Series clarification

  1. #1
    Junior Member Tclack's Avatar
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    Maclaurin/Taylor/Power Series clarification

    Am I right in thinking that a Maclaurin or less specifically Taylor series is the infinite sum representation of a function?

    So If I want to calculate e^5 exactly and I had infinite time and patience, that If I calculated out:
     1+5+\frac{5^2}{2!}+\frac{5^3}{3!}+... That I would arrive at the value for e^5?

    If so
    , then I've seen it works with trig functions, log functions and x to the power of some changing variable. It also works for fractions of all of those... Are there any limitations? Or can ANY function be represented as a Maclaurin or Taylor series
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    MHF Contributor
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    The function needs to be defined and infinitely differentiable at the point that it is centred.
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    Grand Panjandrum
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    Quote Originally Posted by Prove It View Post
    The function needs to be defined and infinitely differentiable at the point that it is centred.
    You mean like:

    f(x)=e^{-1/x^2}

    around  x=0 ???

    CB
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    Quote Originally Posted by CaptainBlack View Post
    You mean like:

    f(x)=e^{-1/x^2}

    around  x=0 ???

    CB
    Is that defined at x = 0?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Prove It View Post
    Is that defined at x = 0?
    By continuity it may be defined to be zero there.

    CB
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    Hmmm, well is it differentiable at x = 0 in that case?
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    Grand Panjandrum
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    Quote Originally Posted by Prove It View Post
    Hmmm, well is it differentiable at x = 0 in that case?
    You should be able to determine that yourself, but I will tell you: It has derivatives of all orders at x=0, all of them zero.

    (the Taylor series around zero has zero radius of convergence)

    CB
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  8. #8
    Junior Member Tclack's Avatar
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    you said infinitely differentiable and Defined..... What did you mean by 'Defined'? And how would you know if something is infinitely differentiable?
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  9. #9
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    Quote Originally Posted by Tclack View Post
    you said infinitely differentiable and Defined..... What did you mean by 'Defined'? And how would you know if something is infinitely differentiable?
    You would have some basic knowledge about the family of functions you'll be dealing with. It's all to do with "experience".

    When I say defined, I mean x = a needs to be in the domain of your function.


    E.g. you can't have a Taylor series for \ln{x} centred at x = 0, because \ln{0} is not defined.

    You CAN however have it centred around x = 1, because \ln{1} IS defined.
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