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Given a particular expansion, find the function in powers of x

  1. Sep 26, 2011 #1
    I am given this expansion:

    x + 2^2 x^2 + 3^2 x^3 + 4^2 x^4 + ... + k^2 x^k + ...

    To get the function that yields this expansion, I cannot figure out the manipulation to (x)/(1-x)
     
  2. jcsd
  3. Sep 27, 2011 #2

    HallsofIvy

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    That's probably because it is not true. To take a simple example, if x= 1/2, that sum becomes [itex](1/2)+ 4(1/4)+ 9(1/8)+ 16(1/16)+ ...[/itex]
    which is already larger than 1/2+ 1+ 9/8+ 1= 3 and 5/8. Since every term is positive, the sum (if it converges) is larger than 3 and 5/8. But (1/2)(1- 1/2)= (1/2)/(1/2)= 1.

    In fact, 1/(1- x) is the sum of the geometric series [itex]\sum_{n=0}^\infty x^n[/itex] so x/(1- x) is [itex]\sum_{n=0}^\infty x^{n+1}[/itex].
     
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