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Sum of a series

  1. Mar 29, 2005 #1

    cepheid

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    Does anyone have tips on how to sum the following series?

    [tex] \sum_{n=1}^{\infty} n^2 w^n [/tex]

    Region of convergence is for |w| < 1
     
  2. jcsd
  3. Mar 29, 2005 #2

    Galileo

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    Try to exploit it's similarity with a geometric series.
    Hint: Differentiate.
     
  4. Mar 30, 2005 #3

    xanthym

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    [tex] (1) \ \ \ \ z \ = \ \sum_{n=0}^{\infty} w^{n} \ = \ (1 \ - \ w)^{-1} \ = [/tex]

    [tex] (2) \ \ \ \ \ \ \ \ \ \ = \ 1 \ + \ w \ + \ \sum_{n=2}^{\infty} w^{n} [/tex]

    [tex] 3 \ \ \ \ \ \ \frac {dz} {dw} \ = \ \left ( 1 \ - \ w \right )^{-2} = [/tex]

    [tex] (4) \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ 1 \ \ + \ \ \sum_{n=2}^{\infty} n \cdot w^{n-1} \ = \ 1 \ \ + \ \ w^{-1} \cdot \sum_{n=2}^{\infty} n \cdot w^{n} [/tex]

    [tex] (5) \ \ \ \ \ \Longrightarrow \ \sum_{n=2}^{\infty} n \cdot w^{n} \ = \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right) [/tex]

    [tex] 6 \ \ \ \ \ \ \frac {d^{2}z} {dw^{2}} \ = \ 2 \cdot \left ( 1 \ - \ w \right )^{-3} \ = [/tex]

    [tex] (7) \ \ \ \ \ \ \ \ \ \ = \ \sum_{n=2}^{\infty} n \cdot ( n \ - \ 1 ) \cdot w^{n-2} \ = [/tex]

    [tex] (8) \ \ \ \ \ \ \ \ \ \ = \ w^{-2} \cdot \sum_{n=2}^{\infty} n \cdot ( n \ - \ 1 ) \cdot w^{n} \ = [/tex]

    [tex] (9) \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ w^{-2} \cdot \left ( \sum_{n=2}^{\infty} n^{2} \cdot w^{n} \ - \ \sum_{n=2}^{\infty} n \cdot w^{n} \right ) \ = [/tex]

    [tex] (10) \ \ \ \ \ \ \ \ \ \ \ = \ w^{-2} \cdot \left ( \sum_{n=2}^{\infty} n^{2} \cdot w^{n} \ - \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right) \right ) \ [/tex]

    [tex] (11) \ \ \ \ \color{red} \Longrightarrow \ \sum_{n=2}^{\infty} n^{2} \cdot w^{n} \ = \ 2 \cdot w^{2} \cdot \left ( 1 \ - \ w \right )^{-3} \ \ + \ \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right) [/tex]

    [tex] (12) \ \ \ \ \color{red} \Longrightarrow \ \sum_{n=1}^{\infty} n^{2} \cdot w^{n} \ \ = \ \ w \ \ + \ \ 2 \cdot w^{2} \cdot \left ( 1 \ - \ w \right )^{-3} \ \ + \ \ w \cdot \left( \left(1 \ - \ w \right)^{-2} \ - \ 1 \ \right) [/tex]



    ~~
     
    Last edited: Mar 30, 2005
  5. Mar 30, 2005 #4
    Just Wanted To Know How To Make The Summation And Powers
     
  6. Mar 30, 2005 #5

    xanthym

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    Try the following URL. Click on the actual formula or graphic for a pop-up window showing the "tex" code.
    https://www.physicsforums.com/showthread.php?t=8997


    ~~
     
    Last edited: Mar 30, 2005
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