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Sum of a power series - help!

  1. Jan 2, 2013 #1
    1. The problem statement, all variables and given/known data
    I am trying to find the sum of the series in the attachment.


    2. Relevant equations



    3. The attempt at a solution
    I have tried to use various series and their derivatives, to not much avail.
    I am not sure how to handle the n^2 factor.
    Should I break it down to two series?
    Any suggestions?
     

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  3. Jan 2, 2013 #2

    lurflurf

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    You know about derivatives? Good do this

    [tex]\sum_{k=1}^\infty k^2 x^{n-1}=\left( x \left( x \sum_{k=1}^\infty x^{k-1}\right)^\prime \right)^\prime=\left( x \left( x \frac{1}{1-x}\right)^\prime \right)^\prime[/tex]

    |x|<1
    your case will be x=1/10
     
  4. Jan 2, 2013 #3
    If I am not mistaken, this yields 700/729, which, according to Wolfram, is incorrect. Would you please account for that?
     
  5. Jan 2, 2013 #4

    Dick

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    You are mistaken. Check it again.
     
  6. Jan 2, 2013 #5
    Would you please explain how it was arrived at?
     
  7. Jan 2, 2013 #6

    Dick

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    Basically you take x^(k-1). Multiplying by x and differentiating gives you k*x^(k-1). Doing the same thing again gives k^2*x^(k-1). Which is the form you want. Now sum the initial x^(k-1) as a geometric series and repeat the same sequence of operations on the function of you get.
     
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