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Math Help - Help with order

  1. #1
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    Help with order

    Why is it possible to represent sin(x) = x + O(x^3)
    thanks for any help
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  2. #2
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    Quote Originally Posted by hmmmm View Post
    Why is it possible to represent sin(x) = x + O(x^3)
    thanks for any help
    Look at the McLaurin series for sin(x): \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1}= x- \frac{1}{6}x^3+ \cdot\cdot\cdot.
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  3. #3
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    this is where the problem arose for me because there are x^5 and so on terms in the expansion how can these all be expressed as O(x^3) or is this just an estimation??
    thankyou for the help and any future help
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    Quote Originally Posted by hmmmm View Post
    this is where the problem arose for me because there are x^5 and so on terms in the expansion how can these all be expressed as O(x^3) or is this just an estimation??
    thankyou for the help and any future help

    \sin x = x + O(x^3) \Longleftrightarrow |\sin x - x| \leq M|x^3|\,\,\,as\,\, x \rightarrow 0\,\,\mbox{ and for some constant M}

    Using now the MacClaurin series of \sin x that HallsofIvy wrote we get \sin x -x=-\frac{x^3}{6}+\frac{x^5}{120}-... , and
    this is less than some constant times x^3 when x is close enough to zero (can you see why?)

    Tonio
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