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Math Help - Taylors Series Confusion

  1. #1
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    Taylors Series Confusion

    Hello, just thought i'd quickly say, I'm new here =].

    I'm doing some problems on Taylors Series, like this one:

    Find the Taylor polynomial of order 2 at x = a generated by the
    function f(x) = sin(sin x) at a = 0:

    I did this, coming to the correct answer by finding the 2 derivatives etc. The solutions however gave me this, which was a lot difference from my working:

    sin(sin x) = sin x + O(sin^3 x) = x + O(x^3);
    Hence P2(x) = x:

    I was hoping somebody could perhaps explain the O symbol to me? and how sin(sin x) = sin c+ O(sin^3 x) etc.

    Another is:

    Verify that (sin x)/x = 1 - (x^2)/6 + o(x^3)

    Thanks =]
    Last edited by Redtwit; April 28th 2011 at 09:46 AM.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Redtwit View Post
    Hello, just thought i'd quickly say, I'm new here =].

    I'm doing some problems on Taylors Series, like this one:

    Find the Taylor polynomial of order 2 at x = a generated by the
    function f(x) = sin(sin x) at a = 0:

    I did this, coming to the correct answer by finding the 2 derivatives etc. The solutions however gave me this, which was a lot difference from my working:

    sin(sin x) = sin x + O(sin^3 x) = x + O(x^3);
    Hence P2(x) = x:

    I was hoping somebody could perhaps explain the O symbol to me? and how sin(sin x) = sin c+ O(sin^3 x) etc.
    Use the indentity...

    (1)

    ... and then find the McLaurin expansion of each exponential in (1) proceeding as in...

    http://www.mathhelpforum.com/math-he...nz-178887.html

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
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    near Piacenza (Italy)
    Posts
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    Quote Originally Posted by Redtwit View Post
    Hello, just thought i'd quickly say, I'm new here =].

    Verify that (sin x)/x = 1 - (x^2)/6 + o(x^3)
    Starting from the well known...

    (1)

    ... deviding both terms by x You obtain...

    (2)

    Kind regards

    \chi \sigma
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