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Math Help - Taylor + Power Series

  1. #1
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    Taylor + Power Series

    I was having some trouble with these:

    1. Find the Taylor polynomial of degree 4 centered at c=2 for the function f(x)=\sqrt[3]{x}

    2. Given e \approx 1 + 1 + {1^2\over2!}+ {1^3\over3!}+ {1^4\over4!}+ {1^4\over4!}, use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error.

    3. Find the radius of convergence of the power series \sum_{n=0}^{\infty}{(2n)!x^{2n} \over {n!}}

    Thank you and sorry for all the questions!
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  2. #2
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    Quote Originally Posted by bakanagaijin View Post
    2. Given e \approx 1 + 1 + {1^2\over2!}+ {1^3\over3!}+ {1^4\over4!}+ {1^4\over4!}, use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error.
    Let T_5(1) be the above Taylor polynomial.
    We have e^1 - T_5(1) = \frac{e^y}{6!}(1)^6 where 0<y<1. Note that e^y < 3.
    Thus, |e - T_5(1)| \leq \frac{3}{6!} .
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  3. #3
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    Quote Originally Posted by bakanagaijin View Post
    3. Find the radius of convergence of the power series \sum_{n=0}^{\infty}{(2n)!x^{2n} \over {n!}}
    Ratio test,
    \frac{(2n+2)!|x|^{2n+2}}{(n+1)!}\cdot \frac{n!}{(2n)!|x|^{2n} } = (2n+1)|x|^2
    This never converges unless x=0.
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