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Homework Help: Radius of convergence

  1. Apr 13, 2010 #1
    1. The problem statement, all variables and given/known data

    find the radius of convergence and interval of convergence of the series


    Σ (3x-2)^2 / n 3^n
    n=1


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 13, 2010 #2
    You need to clean up your notation a little bit, but the ROC for this series is easily found using any number of tests. Try the ratio test or the limsup test.
     
  4. Apr 14, 2010 #3

    HallsofIvy

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    Science Advisor

    [tex]\sum_{n=1}^\infty \frac{(3x-2)^n}{n n!}[/tex]
    is, I think, what you want.

    Ratio test is probably best.
    [tex]a_n= \frac{|3x-2|^n}{n n!}[/tex]
    and
    [tex]a_{n+1}= \frac{|3x-2|^{n+1}}{(n+1) (n+1)!}[/tex]

    so the ratio is
    [tex]\frac{|3x-2|^{n+1}}{(n+1)(n+1)!}\frac{n n!}{|3x-2|^n}= |3x-2|\frac{n}{(n+1)^2}[/tex]
    What is the limit of that as n goes to infinity? If that is less than 1, the series will converge.
     
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