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Sum of geometric series

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  1. Jan 22, 2015 #1
    1. The problem statement, all variables and given/known data

    Find the sum of the following series: Σ n*(1/2)^n (from n = 1 to n = inf).

    2. Relevant equations

    I know that Σ r^n (from n = 0 to n = inf) = 1 / (1 - r) if |r| < 1.

    3. The attempt at a solution

    I began by rescaling the sum, i.e.

    Σ (n+1)*(1/2)^(n+1) (from n = 0 to n = inf)
    = Σ (n+1)*(1/2)*(1/2)^n (from n = 0 to n = inf)
    = (1/2) ∑ (n+1)*(1/2)^n (from n = 0 to n = inf)
    = (1/2) ( ∑ n*(1/2)^n (from n = 0 to n = inf) + ∑ (1/2)^n (from n = 0 to n = inf) )
    = (1/2) ( ∑ n*(1/2)^n (from n = 0 to n = inf) + 1 / (1 - 1/2) )
    = (1/2) ( ∑ n*(1/2)^n (from n = 0 to n = inf) + 2 )

    I'm stuck here. I don't know how to evaluate the first summation. Any help would be appreciated!
     
  2. jcsd
  3. Jan 22, 2015 #2

    Dick

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    Try taking the derivative with respect to r of both sides of Σ r^n (from n = 0 to n = inf) = 1 / (1 - r). Can you relate that result to your problem?
     
  4. Jan 22, 2015 #3

    LCKurtz

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    Hint: You know ##s = \sum r^n##. Think about differentiating that with respect to ##r## and working with that.
     
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