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Math Help - taylor series problem

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    taylor series problem

    Given a function f such that f(3)= 1 and f^{(n)}(3)=\frac{(-1)^n n!}{(2n+1)2^n}

    (a) write the first four nonzero terms and the general term of the Taylor series for f around x= 3

    (b) find the radius of convergence of the Taylor series

    (c) show that the third degree Taylor polynomial approximates f(4) within 0.01
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    Quote Originally Posted by yoman360 View Post
    Given a function f such that f(3)= 1 and f^{(n)}(3)=\frac{(-1)^n n!}{(2n+1)2^n}

    (a) write the first four nonzero terms and the general term of the Taylor series for f around x= 3

    (b) find the radius of convergence of the Taylor series

    (c) show that the third degree Taylor polynomial approximates f(4) within 0.01
    (a) You should know that

    f(x) = \sum_{n = 0}^{\infty}\frac{f^{(n)}(a)}{n!}(x - a)^n.

    In this case, a = 3.

    Substitute all your known information, then write the first four nonzero terms.


    (b) Find \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right|.

    The series converges when this limit is less than 1.

    So solve \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right| < 1 for x.
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