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Math Help - Maclaurin Series with arctan

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    Maclaurin Series with arctan

    Find a Maclaurin series for f(x) = (arctan(x^3))/(x^3)



    any help appreciated
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    Quote Originally Posted by Grimey View Post
    Find a Maclaurin series for f(x) = (arctan(x^3))/(x^3)



    any help appreciated
    Build this up in stages, starting with

    \frac1{1+x^2} = 1-x^2+x^4-x^6+\ldots (binomial series).

    Integrate both sides to get

    \arctan x = x - \frac{x^3}3 + \frac{x^5}5 - \frac{x^7}7 + \ldots.

    Divide both sides by x:

    \frac{\arctan x}x = 1 - \frac{x^2}3 + \frac{x^4}5 - \frac{x^6}7 + \ldots.

    Finally, replace x by x^3:

    \frac{\arctan x^3}{x^3} = 1 - \frac{x^6}3 + \frac{x^{12}}5 - \frac{x^{18}}7 + \ldots.
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