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Homework Help: Help on infinite series

  1. Sep 5, 2010 #1
    1. The problem statement, all variables and given/known data

    I want to find a closed form formula for:

    [tex]x+2x^2+3x^3+4x^4+\ldots[/tex]

    I know that this can be written as:

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

    but I would like to have a closed formula for this.

    The formula for an infinite geometric series is:
    [tex]\sum_{n=0}^{\infty}x^n = \frac{1}{1-x}[/tex]

    Which is somewhat close but the series in question is not exactly geometric.

    How do I go about doing this?
     
  2. jcsd
  3. Sep 5, 2010 #2

    rock.freak667

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    Homework Helper

    Try differentiating the formula for the geometric series.
     
  4. Sep 5, 2010 #3

    Hurkyl

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    Also, the usual "trick" for deriving geometric series also works for that one -- combine the original series S with the series xS to produce something simpler.
     
  5. Sep 6, 2010 #4
    this sort of series is called an arithmetic geometric progression (AGP) or something....like someone said, multiply by x and then subtract to get a simple geometric progression....in this case you could even divide by x
     
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