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Math Help - Taylor Expansion

  1. #1
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    Taylor Expansion

    What is the Taylor Expansion of:

    sin(sinx) at x=0 up to terms of order 3 ?

    I know that the Taylor of sin(x) is: x - x^3/3 + ...

    But then what is the Taylor of sin(x - x^3/3 + ...) ?

    Thanks.
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  2. #2
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    Quote Originally Posted by jzellt View Post
    What is the Taylor Expansion of:

    sin(sinx) at x=0 up to terms of order 3 ?

    I know that the Taylor of sin(x) is: x - x^3/3 + ...

    But then what is the Taylor of sin(x - x^3/3 + ...) ?

    Thanks.
    \sin{X} = X - \frac{X^3}{3!} + \frac{X^5}{5!} - \frac{X^7}{7!} + \dots - \dots.

    In your case, X = \sin{x}.

    So \sin{(\sin{x})}

     = \left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)

     - \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^3}{3!}

     + \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^5}{5!}

     - \frac{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots - \dots\right)^7}{7!} + \dots - \dots


    Very long I know, and some simplifying needs doing as well...
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  3. #3
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    The fact that you only need "to order 3" simplifies a lot!
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