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Math Help - A Series question

  1. #1
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    A Series question

    Let f (x) = (x -arctanx) /(x ^3 )for each x ≠ 0 Find the power series with sum equal to f (x) (when | x | ≤ 1, x ≠ 0), and use this power series to calculate the limit limx → 0f (x).
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  2. #2
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    Re: A Series question

    What have you tried? Do you know the power series for \arctan x? Have you tried just plugging that in and simplifying?
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    Re: A Series question

    Yes i did, i got series (-1)^(n+1) *x^(2n-2) (series from n=1 to infinity)
    But it was not the correct answer

    The correctly answer is ∑k=0∞((−1)^k)/(2k+3)*x^(2k)

    1/3

    I do not understand how he get this answer. Have you any suggestion?
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  4. #4
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    Re: A Series question

    x-\arctan x = x - \sum_{n=0}^\infty (-1)^n \dfrac{x^{2n+1}}{2n+1}

    The first term of the \arctan x series is x. So, when you perform the subtraction, you get:

    x-\arctan x = \sum_{n=0}^\infty (-1)^n \dfrac{x^{2n+3}}{2n+3}

    Then, dividing by x^3 gives the series

    \dfrac{x-\arctan x}{x^3} = \sum_{n=0}^\infty (-1)^n \dfrac{x^{2n}}{2n+3}
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  5. #5
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    Re: A Series question

    Thank you. It is the same i got, but i should reindeks the series to start from n=0 .

    I am leserinnlegg Latex and i use texmaker. How i can write the dokument in latex in this forum. Can you help?
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  6. #6
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    Re: A Series question

    Quote Originally Posted by Haytham1111 View Post
    Thank you. It is the same i got, but i should reindeks the series to start from n=0 .

    I am leserinnlegg Latex and i use texmaker. How i can write the dokument in latex in this forum. Can you help?
    Click Reply With Quote on any post with LaTeX to see how it is formatted on this forum.
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