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Evaluation of power series

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data

    evaluate ∑ n^2.x^n where 0<x<1

    2. Relevant equations



    3. The attempt at a solution
    let a_n = n^2 and c=0
    then radius of convergence, R=1
    hence the series convergences when |x|<1
    let f(x) = ∑ n^2.x^n
    then f'(x) = ∑ n^3.x^n-1 for n=0 to infinity
    then f'(x) = ∑ (n+1)^3.x^n for n=1 to infinity

    from here, how to I derive a function ∑ (n+1)^3.x^n so as to integrate it to get the sum?
     
  2. jcsd
  3. Mar 3, 2010 #2

    tiny-tim

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    Hi mathmathmad! :smile:

    (have an infinity: ∞ and try using the X2 and X2 tags just above the Reply box :wink:)
    Why are you making it more complicated? :redface:

    Hint: try integrating. :wink:
     
  4. Mar 3, 2010 #3
    differentiating is a much better option fyi
     
  5. Mar 3, 2010 #4
    It's somewhat hard to start from your f(x) and find a series you know.

    Instead, try starting from a series you know and apply these methods to get f(x).
     
  6. Mar 3, 2010 #5
    what to integrate?
    intergrate n^2.x^n?
     
  7. Mar 3, 2010 #6

    tiny-tim

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    Sort-of.

    Suppose it was ∑ nxn … what would you integrate? :wink:
     
  8. Mar 3, 2010 #7
    I still say (expanding a little on my hint) to start with an expression for

    ∑ xn

    and try to derive an expression for your series.
     
  9. Mar 4, 2010 #8

    tiny-tim

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    Hi Mathnerdmo! :smile:
    ah, i see what you mean now …

    your method is basically the same as mine, but in reverse …

    i'm integrating the question to try to get something easier, while you're starting with something easier, and differentiating to get the question. :wink:

    Yes, if mathmathmad wants to start with ∑ xn and differentiate it, that's fine. :smile:
     
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