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Homework Help: Infinite series in terms of x

  1. Mar 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Hi,

    How do i determine de result in terms of x of this series for x < 1:

    (Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)

    Thanks


    3. The attempt at a solution

    I know that (Sum(i=0..+infinity;x^i) will tend do 1/(1-x) but i don't know what the numerator will tend to

    Thanks
     
  2. jcsd
  3. Mar 23, 2010 #2

    tiny-tim

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    Homework Helper

    Hi benf.stokes! :smile:

    (have a sigma: ∑ and an infinity: ∞ and try using the X2 and X2 tags just above the Reply box :wink:)
    Hint: integrate. :smile:
     
  4. Mar 29, 2010 #3
    Thanks, I figured it out but by differentiating (sorry for the delay but I was netless for a few days):

    [tex]
    \sum_{n=0}^{\infty}x^n=\frac{1}{1-x}
    [/tex]

    [tex]
    \sum_{n=0}^{\infty}nx^{n-1}=\frac{1}{(1-x)^2}
    [/tex]

    [tex]
    \sum_{n=0}^{\infty}nx^n=\frac{x}{(1-x)^2}
    [/tex]

    How would it be done by integrating? The other way around?
     
    Last edited: Mar 29, 2010
  5. Mar 29, 2010 #4

    tiny-tim

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    Sum the integral, and then differentiate that sum. :smile:
     
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