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Finding the radius of convergence of a series.

  1. May 1, 2012 #1
    1. The problem statement, all variables and given/known data

    What is the radius of convergence of the Taylor Series of the function [itex]f(z) = z cot(z)[/itex], at the point [itex]z = 0[/itex]?

    2. Relevant equations

    Taylor series is given by:
    [tex]\sum_{k=0}^{\infty} \frac{f^{(k)}(z_{0})}{k!} (z - z_{0})[/tex]

    And the radius R by:
    [tex]\lim_{n \to \infty} |\frac{a_{n}}{a_{n+1}}|[/tex]


    3. The attempt at a solution

    The problem here is to find a pattern to represent the function as a series.
    I did some derivatives and tried to substitute [itex]cot(z)[/itex] for it's representation as an exponential, but all I've got is division by zero.

    [tex]f(z) = z cot(z) = z (\frac{e^{2z} + 1}{e^{2z} -1})[/tex]

    [tex]\frac{d f}{dz} = \frac{-e^{4z} + 1 + 4 z e^{2*z}}{(e^{2z}-1)^2}[/tex]
    And so on...
    At the point z = 0, the function blows up.

    As a trigonometrical representation, it's the same thing.
    [tex]\frac{df(z)}{dz} = cot(z) - z csc(z)[/tex]


    I'm not really going anywhere here.
    Please help.
     
  2. jcsd
  3. May 1, 2012 #2
    Maybe the convergence radius is zero ?
     
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