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Math Help - Taylor Polynomials

  1. #1
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    Taylor Polynomials

    I missed the class when my professor went over this. Can anyone show how this is to be done?

    Produce a general formula for the degree n Taylor polynomials using a = 0 as the point of approximation.

    a) 1 / (1-x)

    b) Square Root of (1+x)

    Thanks for the help!
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  2. #2
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    Quote Originally Posted by jzellt View Post
    I missed the class when my professor went over this. Can anyone show how this is to be done?

    Produce a general formula for the degree n Taylor polynomials using a = 0 as the point of approximation.

    a) 1 / (1-x)

    b) Square Root of (1+x)

    Thanks for the help!
    Assume that \frac{1}{1 - x} can be expressed as a polynomial.

    Therefore

    \frac{1}{1 - x} = a + bx + cx^2 + dx^3 + ex^4 + \dots.

    To find a, let x = 0.


    So 1 = a.


    To find b, take the derivative of both sides, then set x = 0.

    So \frac{1}{(1 - x)^2} = b + 2cx + 3dx^2 + 4ex^3 + 5fx^4 + \dots

    1 = b


    To find c, take the derivative of both sides, then set x = 0.


    So \frac{2}{(1 - x)^3} = 2c + 6dx + 12ex^2 + 20fx^3 + 30gx^4 + \dots

    2 = 2c

    1 = c.


    To find d, take the derivative of both sides, then set x = 0

    So \frac{6}{(1 - x)^4} = 6d + 24ex + 60fx^2 + 120gx^3 + \dots

    6 = 6d

    1 = d.



    I think it's relatively easy to see that all the coefficients will be equal to 1.


    So \frac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + \dots.


    We can also realise this using the fact that for a convergent infinite geometric series

    S_{\infty} = \frac{a}{1 - r} provided |r| < 1.


    If we take a = 1 and r = x, we find

    \frac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + \dots, provided that |x| < 1.
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  3. #3
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    Quote Originally Posted by jzellt View Post
    I missed the class when my professor went over this. Can anyone show how this is to be done?

    Produce a general formula for the degree n Taylor polynomials using a = 0 as the point of approximation.

    a) 1 / (1-x)

    b) Square Root of (1+x)

    Thanks for the help!
    See the Wikipedia article on Taylor series, then degree n Taylor polynomial is the series truncated at the degree n term.

    CB
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