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Taylor series error term - graphical representation

  1. Nov 3, 2012 #1
    Hello all,
    Recently I've found something very interesting concerning Taylor series.
    It's a graphical representation of a second order error bound of the series.
    Here is the link: http://www.karlscalculus.org/l8_4-1.html [Broken]

    My question is: is it possible to represent higher order error bounds in a similar way?
    For example: third order error term would have "3! = 6" in a denominator...
    I know that Taylor series is based on Mean Value Theorem and I know the proof of it.
    However it would become much clearer if it was possible to represent error bounds in a graphical way.

    Have a nice weekend.
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Nov 3, 2012 #2


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    Homework Helper

    Yes, see any calculus book, or this


    It follows from the mean value theorem

    Often in simple examples the error is well approximated by the next term as in

    [tex]\sin(x) \sim \sum_{k=0}^\infty \frac{x^k}{k!} \sin\left(
    x+k \frac{\pi}{2}\right)[/tex]
    Last edited: Nov 3, 2012
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