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Find the sum

  1. Mar 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the sum [tex]\sum_{0}^{\infty} \frac {n^2} {3^n}[/tex]

    2. Relevant equations
    3. The attempt at a solution

    I don't know how to go about finding this sum, I have a guess of what it will be just by adding the first ten terms or so, but how do I find an actual approximation?
     
  2. jcsd
  3. Mar 11, 2009 #2
    Try multiplying three geometric series together and see how close it is to the above series. I think it's simmilar to repeated roots in differential equations.
     
  4. Mar 11, 2009 #3
    The only similar series I see here are:

    [tex]\sum_{0}^{\infty} (\frac {1} {3})^n[/tex]

    [tex] \sum_{0}^{\infty} n^2[/tex]
     
  5. Mar 11, 2009 #4
    Try looking at the second derivative of [tex]\sum_0^\infty x^n[/tex].
     
  6. Mar 11, 2009 #5

    lurflurf

    User Avatar
    Homework Helper

    homogeneous differentiation

    sum=[(xD)^2](1/(1-x))|x=1/3

    that is
    sum=g(1/3)
    when
    f(x)=1/(1-x)
    and
    g(x)=x[x*f'(x)]'=(x^2)*f''(x)+x*f'(x)
     
    Last edited: Mar 11, 2009
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