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Math Help - Proof of Theorem involving Maclaurin Polynomials

  1. #1
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    Proof of Theorem involving Maclaurin Polynomials

    I'm trying to understand the proof found here:

    Calculus 9th edition, chapter 9, section 7, exercise 71
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    The theorem is:
    ------------------------------------------------

    If f is an even function, then it's nth Maclaurin polynomial contains only terms with even powers of x.

    ------------------------------------------------

    I don't understand why the coefficients of the terms are zero for the odd derivatives.
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  2. #2
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    Which part of the proof do you get hung up on?
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  3. #3
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    Quote Originally Posted by Ackbeet View Post
    Which part of the proof do you get hung up on?
    I don't understand why the coefficients of the terms are zero for the odd derivatives.
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  4. #4
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    Take a MacLaurin series: f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}\,x^{2}+\dots

    If you take the first derivative of this, you're going to make the constant term disappear, and you'll get f'(0)+f''(0)x+\dots

    If the f'(0) is not zero, then in general, you can't have f'(x) be an odd function (which it must be, since it is the derivative of an even function). Get the idea?
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