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Math Help - Disk of Convergence

  1. #1
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    Disk of Convergence

    (a) Determine the disk of convergence of the power series

    (n^5)(z-3i)^n/4^n) for n=1 to infinity

    (b) Use taylore theorem to determine the taylor series about 2i for the functio f(z) = Log(z+i), giving an expression for the general term of teh series. Also, state the largest open disk on which the function f is represented by this taylor series.

    (c) Find the taylor series about 0 for each of the following functions:

    (i) f(z) = (e^2z)Coshz (up to the terms in z^3)
    (ii) f(z) - Log(2-cosz) (up to the terms in z^6)

    (d) Let the function f be entire and suppose that

    f(i/n) = -(2/n^3) for all n subset N

    show that f(z) = - 2iz^3

    Thankyou
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  2. #2
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    Quote Originally Posted by moolimanj View Post
    (a) Determine the disk of convergence of the power series

    (n^5)(z-3i)^n/4^n) for n=1 to infinity
    We have,
    \sum_{n=1}^{\infty} \frac{n^5(z-3i)^n}{4^n}

    Use the ratio test,
    \left| \frac{n^5 (z-3i)^n}{4^n} \right|^{1/n} = \frac{(n^{1/n})^5 |z-3i| }{4}

    To evaluate the limit of this sequence we use the famous limit \lim n^{1/n}=1.

    Thus, (for absolute convergence we make it less than 1),
    \frac{|z-3i|}{4} <1
    Thus,
    |z-3i| < 4
    Hence the open disk is described as,
    N(3i,4)
    I do not think any points on the boundary of the disk converge.
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