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    Taylor Series, Radius of Convergence

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    Let f(x)=ln?(1+x).

    (a) Prove by induction that for n ≥ 1,
    f^(n) (x) = (-1)^(n-1) ((n-1)!/(1+x)^n ))
    (It is not necessary here to determine the derivatives from first principles)

    (b) Write down the Taylor series for f at a = 0.

    (c) What is the radius of convergence of the power series in (b)?

    (d) Use Taylor polynomial of degree 5 to give an approximation to ln?(2).

    (e) Use the error term from Taylor's Theorem to show that the error in (d) is at most 1/6. (Note that this is still not a particularly good approximation.)

    See attached document for equation.

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    Solution Summary

    Analyzes the Taylor Series and radius of convergence. The principle of mathematical induction is also incorporated.