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Confused about taylor approximation

  1. Apr 18, 2004 #1
    I am a bit confused about taylor approximation. Taylor around [tex]x_0[/tex] yields
    [tex]
    f(x) = f(x_0) + f'(x_0)(x-x_0) + O(x^2)
    [/tex]

    which is the tangent of f in [tex]x_0[/tex], where
    [tex]
    f'(x) = f'(x_0) + f''(x_0)(x-x_0) + O(x^2)
    [/tex]

    which adds up to
    [tex]
    f(x) &=& f(x_0) + (f'(x_0) + f''(x_0)(x-x_0) + O(x^2))(x-x_0)+O(x^2) \\ &=& f(x_0) + f'(x_0)(x-x_0) + f''(x_0)(x-x_0)^2 + O(x^3)
    [/tex]
    But it should be
    [tex]
    f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + O(x^3)
    [/tex]

    Where does the 2! come from? Is this approach completely incorrect?
     
  2. jcsd
  3. Apr 18, 2004 #2

    arildno

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    Dearly Missed

    You have ignored in line 3 the O(x^(2))-term from the expansion of f(x).
    Hence line 3 is not accurate to O(x^(3)), it's only accurate to O(x^(2)).
     
  4. Apr 18, 2004 #3

    Hurkyl

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    Do you remember how to derive, from the limit definition of the derivative, the differential approximation formula:

    [tex]
    f(x+\epsilon ) = f(x) + \epsilon f'(x) + \epsilon \delta(x, \epsilon)
    [/tex]

    Where [itex]\lim_{\epsilon \rightarrow 0} \delta(x, \epsilon) = 0[/itex]?

    Try writing the second derivative with limits, and see if any approach suggests itself.
     
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