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Math Help - taylors series, complex variables

  1. #1
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    taylors series, complex variables

    expand the function f(z)=1/z^2 in a taylors series around z=1. What is the radius of convergence?


    Attempted soln: Since 1/z^2 has a singularity at 0, isnt the radius of convergence just R=1-0 ?
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  2. #2
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    Quote Originally Posted by stumped765 View Post
    expand the function f(z)=1/z^2 in a taylors series around z=1. What is the radius of convergence?


    Attempted soln: Since 1/z^2 has a singularity at 0, isnt the radius of convergence just R=1-0 ?

    Well, yes...at most, but it could be less, right? You'll have to check it with the Taylor series directly.

    Tonio
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  3. #3
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    i found the taylor expansion to be
    SUM(n=0 to infinity) (-1)^n (n+1)*(z-1)^n

    does that sound right?
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  4. #4
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    Quote Originally Posted by stumped765 View Post
    i found the taylor expansion to be
    SUM(n=0 to infinity) (-1)^n (n+1)*(z-1)^n

    does that sound right?

    Sounds, looks and is right...and now verify that the conv. radius indeed is 1. Good work.

    Tonio
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  5. #5
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    is there any way to find radius of convergence without using the ratio/comparison tests?
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  6. #6
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    You could do this:

     <br />
f(x) = \frac{1}{z^2} = \frac{1}{1 + (z^2 - 1)}

     \ \ \ \mid R \mid \leq 1

     \ \ \ R = (z^2 - 1)

     \ \ \ \mid (z^2 - 1) \mid \ \  \leq 1<br />
    Last edited by lilaziz1; March 29th 2010 at 04:21 PM.
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