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Math Help - How do i show the Taylor Series of inverse sine?

  1. #1
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    How do i show the Taylor Series of inverse sine?

    Hiya, i've got to show that the Taylor series for arcsine(x) = x + (x^3/6) + (3x^5/40) +....

    I've tried using the general formula where i set it out as a table of derivatives i.e f'(a), f''(a), f'''(a) etc and when a=0 but it doesn't work and i only get x. The rest turns out as 0 for the coeffiecent when i substitute a=0 in to the higher derivatives.

    Does anyone have any ideas? Many thanks.

    Chinkmeista.
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  2. #2
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    Maybe the derivatives were incorrect?
    f(x) = sin^{-1}x
    f(0) = sin^{-1}0 = 0

    f'(x) = \frac{1}{\sqrt{1-x^{2}}}

     f'(0) = 1

    f''(x) = \frac{x}{(1-x^{2})^{\frac{3}{2}}}

    f''(0)= 0

    f'''(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}}

    f'''(0) = 1

    f^{(4)}(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}}

    f^{(4)}(0) = 0

    f^{(5)}(x) = \frac{24x^{4}+12x^{2}+9}{(1-x^{2})^{\frac{9}{2}}}

    f^{(5)}(0) = 9

    This gives \sin^{-1}x = x+\frac{x^{3}}{3!}+\frac{9x^{5}}{5!}+... = x+\frac{x^{3}}{6}+\frac{3x^{5}}{40}+...
    Last edited by xxlvh; November 7th 2009 at 07:56 PM. Reason: LaTex errors
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by chinkmeista View Post
    Hiya, i've got to show that the Taylor series for arcsine(x) = x + (x^3/6) + (3x^5/40) +....

    I've tried using the general formula where i set it out as a table of derivatives i.e f'(a), f''(a), f'''(a) etc and when a=0 but it doesn't work and i only get x. The rest turns out as 0 for the coeffiecent when i substitute a=0 in to the higher derivatives.

    Does anyone have any ideas? Many thanks.

    Chinkmeista.
    It works fine just taking the derivatives, though it is pretty tedious. The third derivative is \frac{2x^2+1}{(x^2-1)^2\sqrt{1-x^2}} and the fifth is \frac{24x^4+72x^2+9}{(x^2-1)^4\sqrt{1-x^2}}.

    Note a pattern: The (n+1) derivative of inverse sine is \frac{(-1)^n(n!x^n+...)}{(x^2-1)^n\sqrt{1-x^2}}

    The numerator is some polynomial, but the only obvious pattern to it was the factorial coefficient for the highest power (though there is undoubtedly some pattern for the rest of the terms).
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  4. #4
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    Quote Originally Posted by xxlvh View Post
    Maybe the derivatives were incorrect?
    f(x) = sin^{-1}x
    f(0) = sin^{-1}0 = 0

    f'(x) = \frac{1}{\sqrt{1-x^{2}}}

     f'(0) = 1

    f''(x) = \frac{x}{(1-x^{2})^{\frac{3}{2}}}

    f''(0)= 0

    f'''(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}}

    f'''(0) = 1

    f^{(4)}(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}}

    f^{(4)}(0) = 0

    f^{(5)}(x) = \frac{24x^{4}+12x^{2}+9}{(1-x^{2})^{\frac{9}{2}}}

    f^{(5)}(0) = 9

    This gives \sin^{-1}x = x+\frac{x^{3}}{3!}+\frac{9x^{5}}{5!}+... = x+\frac{x^{3}}{6}+\frac{3x^{5}}{40}+...
    Ah ok, but how does the 3rd derivative equal the 4th one: f'''(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}} and f^{(4)}(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}} ?

    And also, for the 3rd derivative, how did you get the 2x+1 on top? did you use the quotient rule?
    Last edited by chinkmeista; November 7th 2009 at 09:18 PM. Reason: extra bit
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  5. #5
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by chinkmeista View Post
    Ah ok, but how does the 3rd derivative equal the 4th one: f'''(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}} and f^{(4)}(x) = \frac{2x^{2}+1}{(1-x^{2})^{\frac{5}{2}}} ?
    It's supposed to be \frac{6x^3+9x}{(1-x^2)^{7/2}}, so f^{(4)}(0)=0.
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  6. #6
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    Typing error, my apologies I was quickly trying to recopy those results from a program I used to find the derivatives. However they're still slightly off from the ones that redsoxfan325 has provided so it is best to disregard the ones I posted, it doesn't seem to always work for inverse trig functions..lucky coincidence it resulted with the same coefficients.
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  7. #7
    Super Member redsoxfan325's Avatar
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    I was doing them on my TI-89. The only difference was the numerator for the f^{(5)}(x). Everything else was the same, though we were representing our denominators differently.
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