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Math Help - Rate of Convergence

  1. #1
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    Rate of Convergence

    Hi All,

    I've been trying to figure out how to do the question in the attached file for quite a few days now, and I think I've got most of it figured out. I need to find the rate of convergence of the function as h approaches 0. I have attached both the question and my work. I'm down to the final inequality and have no idea how to find O(h^2) which is the answer to this question.

    The function is lim h->0 (sinh+hcosh)/h

    Any help would be appreciated.

    Thank you.
    Attached Thumbnails Attached Thumbnails Rate of Convergence-img_0050.jpg  
    Last edited by sjmiller; September 30th 2012 at 12:07 PM.
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Rate of Convergence

    Hint:

    Series expansion of each function near 0 is:

    \text{Sin}(x)/x = 1-\frac{x^2}{6}+\frac{x^4}{120}-\frac{x^6}{5040}+O(x)^8 [updated]

    \text{Cos}(x) = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+O(x)^8
    Last edited by MaxJasper; September 30th 2012 at 02:45 PM.
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  3. #3
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    Re: Rate of Convergence

    Quote Originally Posted by MaxJasper View Post
    Hint:

    Series expansion of each function near 0 is:

    \text{Sinh}(x)/x = 1+\frac{x^2}{6}+\frac{x^4}{120}+\frac{x^6}{5040}+O  (x)^8

    \text{Cos}(x) = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+O(x)^8
    Hi, it is sin(x)/x not sinh(x)/x so I'm a bit confused :S
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  4. #4
    Senior Member MaxJasper's Avatar
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    Re: Rate of Convergence

    See update
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