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Math Help - Proof of approximation for function

  1. #1
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    Post Proof of approximation for function

    Hey MathsForum

    If x is small, show that \sqrt{{\left\{\frac{1+x}{1-x}\right\}}} \approx 1 + x + \frac{x^2}{2}

    Can anyone help?

    Thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by dadon View Post
    Hey MathsForum

    If x is small, show that \sqrt{{\left\{\frac{1+x}{1-x}\right\}}} \approx 1 + x + \frac{x^2}{2}

    Can anyone help?

    Thanks
    Method 1 - consider the truncated Taylor expansion of

    \sqrt{{\left\{\frac{1+x}{1-x}\right\}}}

    about 0

    Method 2 - Multiply together the first few terms (and the remainders) of the Taylor series of \sqrt{1+x} and 1/\sqrt{1-x} and discard the terms of order x^3 and higher.

    We have:

    (1+x)^{1/2}=1+(1/2)x+(1/2)(-1/2)x^2/2+O(x^3)

    and:

    (1-x)^{-1/2}=1+(-1/2)(-x)+(-1/2)(-3/2)(-x)^2/2+O(x^3)

    so:

    \sqrt{{\left\{\frac{1+x}{1-x}\right\}}}=1+x+(1/4)x^2-(1/8)x^2+(3/8)x^2+O(x^3) \approx 1+x+x^2/2

    RonL
    Last edited by CaptainBlack; January 27th 2007 at 02:14 PM.
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  3. #3
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    There is another thing you can do but it is similar to what CaptainBlank said. You can parabolize, remeber in Calculus I you did linearization, that is the best line. Here you do the best parabola, which turns out to be the same coefficients as in the Taylor series.
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