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Laurent Series Convergence

  1. Apr 26, 2012 #1
    I am trying to understand the idea of annulus of convergence. This is the example I have been looking at but it has me completely stumped.

    [∞]\sum[/n=1] (z^n!)(1-sin(1/2n))^(n+1)! + [∞]\sum[/n=1] (2n)!/[((n!)^2)(z^3n)]

    All of the examples I have worked on in the past have been complex functions. This one seems odd because it is a Laurent Series.
  2. jcsd
  3. Apr 26, 2012 #2
    If have to go back to Calc II, and find or use a series of test series e.g.

    alternating series
    comparsion tests

    to see how and when they converge...
  4. Apr 27, 2012 #3
    I think your problem stems from evaluating the convergence of the term

    [tex]\sum \frac{b_n}{z^n}[/tex]

    that has a region of convergence "greater" than some number. For example, suppose I let 1/z=w and consider:

    [tex]\sum b_n w^n[/tex]

    and I can use any of the standard tests on that and find out it's radius of convergence is 3. That means




    That gives you the inner radius and the radius of convergence for the other sum gives you the outer radius.
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