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How can i express this Infinite series without a summation symbol?

  1. Apr 23, 2008 #1
    (1/2) + (2/4) + ... + (n/(2^n))

    =

    sum i=1 to i=infinity of (i/(2^i))?


    i know how to express the sum of just 1/(2^i), but not the above

    thanks for the help!
     
  2. jcsd
  3. Apr 23, 2008 #2

    arildno

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    I don't understand the question.
    1. Do you want a symbolic notation for a sum without using the standard summation symbol?
    2. Are you instead asking for how the standard summation symbol looks like?
    3. Do you wish to find an alternate expression for the sum, i.e, calculate it in a manner so that it is easy to evaluate it for an arbitrarily chosen n?
     
  4. Apr 23, 2008 #3

    arildno

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    I think you mean 3, i.e, how to calculate that sum. Am I right?
     
  5. Apr 23, 2008 #4
    yea number 3
     
  6. Apr 23, 2008 #5
    [tex]2^{-n} \left(-n+2^{n+1}-2\right)[/tex]
     
  7. Apr 23, 2008 #6
    This is so instructive...!:rolleyes:
     
  8. Apr 23, 2008 #7

    exk

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    the summation is not 0, the above is only slightly correct ;)
     
  9. Apr 23, 2008 #8
    did he ask for instruction?
     
  10. Apr 24, 2008 #9

    arildno

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    Let us consider the function:
    [tex]F(x)=\sum_{i=1}^{\infty}i*x^{i}[/tex]
    Note that F(1/2) equals your sum!
    Now, we may write:
    [tex]F(x)=x*\sum_{i=1}^{\infty}i*x^{i-1}=x*\frac{d}{dx}\sum_{i=1}^{\infty}x^{i}=x*\frac{d}{dx}(\frac{1}{1-x}-1)=\frac{x}{(1-x)^{2}}[/tex]
    Hence, we easily gain F(1/2)=2.

    For arbitrary finite n, use a similar procedure.
     
  11. Apr 24, 2008 #10
    Bah, I wrote a similar response using generating functions... twice... and physics forums died on me both times so nothing was posted.

    Arildno's response is entirely correct though
     
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