Sum of a Power Series

Date: 02/10/2001 at 18:40:29
From: Ivan
Subject: The sum of a power series

Can you tell me how to calculate the sum of this power series?

  x + 4x^2 + 9x^3 + 16x^4 + ... + n^2x^n + ...    |x|<1

I've found that the series is convergent, but can't get the sum.

Thanks,
Ivan

Date: 02/10/2001 at 19:21:12
From: Doctor Anthony
Subject: Re: The sum of a power series

Multiply the original series by x: 

             S = x + 4x^2 + 9x^3 + 16x^4 + ... +  n^2x^n + ...  
            xS =      x^2 + 4x^3 +  9x^4 + ... + (n-1)^2x^n + ...     

Subtract the second series above from the first:

        (1-x)S = x + 3x^2 + 5x^3 +  7x^4 + ... + (2n-1)x^n + ...
       x(1-x)S =      x^2 + 3x^3 +  5x^4 + ... + (2n-3)x^n + ...

Repeat:

     (1-x)^2*S = x + 2x^2 + 2x^3 +  2x^4 + ...
               = x + 2x^2[1 + x + x^2 + x^3 + ...

Replace the series in brackets with 1/(1-x):

     (1-x)^2*S = x + 2x^2/(1-x)

                    x       2x^2
             S = ------- + -------
                 (1-x)^2   (1-x)^3 

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/