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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

    [tex]\left|\frac{1}{z}\right|<3[/tex]

    or:

    [tex]|z|>1/3[/tex]

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