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Math Help - Taylor Series

  1. #1
    Swaynisha
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    Taylor Series

    Hello,
    I am currently stuck on the following problem in a Calculus BC online course that I am taking while still in high school.

    The first part : The Derivative Form of the Remainder for the Taylor series of f(x) will be denoted by Rsubk(x). Express Rsub5(6) in terms of the value c for the Taylor series for f(x) = sin x when sin x is expanded about a = 0.

    The second part : Use the Derivative Form of the Remainder Rsubk(x) to determine the degree of the Taylor polynomial that approximates sin 6 to within 0.0005 of its actual value.

    Any help whatsoever would be amazing.

    I have been told that the series for f(x)=sin(x) at a=0 is equal to
    tsub(2k+1)(x)= sigma (i=0 to k) of (((-1)^i)/(2i+1)!)*(x^(2i+1))
    But where to go from here?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Swaynisha View Post
    Hello,
    I am currently stuck on the following problem in a Calculus BC online course that I am taking while still in high school.

    The first part : The Derivative Form of the Remainder for the Taylor series of f(x) will be denoted by Rsubk(x). Express Rsub5(6) in terms of the value c for the Taylor series for f(x) = sin x when sin x is expanded about a = 0.
    I know Lagrange's and Cauchy's form for the remainder, but what is the
    derivative form?

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    I know Lagrange's and Cauchy's form for the remainder, but what is the
    derivative form?

    RonL
    OK a bit of research tells us that the "Derivative form" is another name for
    Lagrange's form for the remander.

    So we have:

    f(x) = f(a) + (x-a).f'(a)/1! + (x-a)^2.f"(a)/2! + (x-a)^3.f^{3}(a)/3!

    ........................ +...+ (x-a)^n.f^{n}(a)/n! + R_n


    R_n = (x-a)^{n+1}.f^{n+1}(c)/(n+1)!, for some c between a and x.

    (there is some confusion about if this remainder should have subscript n or n+1
    at least in my mind, that is should n denote the highest power of x in the
    un-ignored part of the expansion, or the number of terms in the expansion)


    RonL
    Last edited by CaptainBlack; May 4th 2007 at 11:12 AM.
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