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Sum of Convergent Series

  1. Sep 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Dear All,

    I have a series that I know to converge but for which I can't work out the infinite sum. It should be something simple.

    [tex]
    S_n = \sum_{j=1}^\infty \beta^j j
    [/tex]

    Can somebody help me with this?

    I think the solution is:

    [tex]
    \frac{\beta}{(1-\beta)^2}
    [/tex]
     
  2. jcsd
  3. Sep 20, 2009 #2
    It looks right:

    [tex] S_n = \beta + 2\beta^2 + 3\beta^3 + \cdots [/tex]

    [tex] \frac{S_n}{\beta} = 1 + 2\beta + 3\beta^2 + \cdots [/tex]

    [tex](\frac{1}{\beta}-1) S_n = 1 + \beta + \beta^2 + \cdots [/tex]

    [tex](\frac{1}{\beta}-1) S_n = \frac{1}{1-\beta}[/tex]

    [tex](\frac{1-\beta}{\beta}) S_n = \frac{1}{1-\beta}[/tex]

    [tex]S_n = \frac{\beta}{(1-\beta)^2}[/tex]

    It's called an arithmetico-geometric series I think,

    [tex]\displaystyle\sum_{n=0}^{\infty}(a+nd)r^n = \frac{a}{1-r} + \frac{rd}{(1-r)^2}[/tex]
     
  4. Sep 20, 2009 #3
    brilliant. I was trying something like that but couldn't get it.

    Thanks for the quick reply.

    Paul
     
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