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Help with summation

  1. Sep 21, 2010 #1
    1. The problem statement, all variables and given/known data
    Evaluate: 1/4+2/16+3/64+4/256+5/1024+.....

    2. Relevant equations

    3. The attempt at a solution
    The summation can be written as: Sum(k=1 to infinity, k/(4^k))
    Then I do not know how to calculate the sum. Please help!
  2. jcsd
  3. Sep 21, 2010 #2


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    You know how to sum x^k/(4^k), right? It's a geometric series. It gives you some function f(x). Now consider what f'(x) is evaluated at x=1.
  4. Sep 21, 2010 #3
    Honestly I do not know how to sum x^k/(4^k), can you explain a little more why we have to put it in a function f(x)? f'(x) at x=1 gonna be (k-ln4)/(4^k) but then ?
  5. Sep 21, 2010 #4


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    The sum of x^k/4^k is geometric because it's the sum of (x/4)^k. Look up the formula for summing a geometric series. The common ratio r=x/4, yes? The result is a function of r, which x/4. So it's a function of x. And when I say f'(x) I mean the derivative with respect to x. Isn't it sum k*x^(k-1)/4^k? No logs needed. Do you see how the k in your sum comes in?
  6. Sep 22, 2010 #5
    Thanks, I got it ^^
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