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Homework Help: Power Series help.

  1. Apr 3, 2014 #1

    STJ

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    1. The problem statement, all variables and given/known data
    Find interval of convergence and radius of convergence of the following infinite series.

    Series from n=1 to infinity ((-3)^n * x^n) / (n*(n)^1/2)


    2. Relevant equations
    Ratio test


    3. The attempt at a solution

    I've started with the ratio test and end up getting 3xn^(3/2) / (n+1)^(3/2) after cancellation. I don't know how to cancel anything else out, I'm guessing L'Hopital's rule but that doesn't seem right. I feel like I should be able to do more cancellation here.
     
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  3. Apr 3, 2014 #2

    Ray Vickson

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    Just use elementary algebra:
    [tex] \frac{n^{3/2}}{(n+1)^{3/2}} = \left( \frac{n}{n+1}\right)^{3/2}[/tex]
    What happens to this ratio when ##n \to \infty?##
     
  4. Apr 3, 2014 #3

    PeroK

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    What can you say about n/(n+1) as n →∞?
     
  5. Apr 3, 2014 #4

    STJ

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    I swear I think to hard sometimes. Thanks.

    And as n/(n+1) n →∞ = 1

    R=1/3, Interval of convergence will be [-1/3, 1/3]
     
  6. Apr 3, 2014 #5

    PeroK

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    Are you sure it converges for x = -1/3?
     
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