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Math Help - Taylor Polynomial

  1. #1
    Del
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    Taylor Polynomial

    Find T(10) (x): the Taylor polynomial of degree 10 of the function f(x)=arctan(x^3) at a=0.

    Can someone show me the steps to finding this, please?
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    Quote Originally Posted by Del View Post
    Find T(10) (x): the Taylor polynomial of degree 10 of the function f(x)=arctan(x^3) at a=0.

    Can someone show me the steps to finding this, please?
    note that if

    f(x)=\tan^{-1}(x) then

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

    by the def of a geometric series we get...

    f'(x)=\sum_{n=0}^{\infty}(-1)^nx^{2n} so if integrate both sides we get

    f(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{2n+1}

    so then f(x^3)=\tan^{-1}(x^3)=\sum_{n=0}^{\infty}\frac{(-1)^n(x^3)^{2n+1}}{2n+1}

    so we get...
    tan^{-1}(x^3)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{6n+3}}{2n+1}

    Just write out as many terms as you need.
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  3. #3
    Del
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    I got this massive polynomial for T(10), but it's incorrect. What am I doing wrong?

    T(10) = [x^3-x^9/3+x^15/5-x^21/7+x^27/9-x^33/11+x^39/13-x^45/15+x^51/17-x^57/19]
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    Quote Originally Posted by Del View Post
    I got this massive polynomial for T(10), but it's incorrect. What am I doing wrong?

    T(10) = [x^3-x^9/3+x^15/5-x^21/7+x^27/9-x^33/11+x^39/13-x^45/15+x^51/17-x^57/19]
    When they say T_{10}(x) They mean a polynomial with degree 10 or less.

    You have a polynomial of degree 57. You would want only the first two terms.

    T_{10}(x)=x^3-\frac{1}{3}x^9 the next term would be degree 15

    Good luck.

    B
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