Power Series from Long Division

Date: 08/31/98 at 20:44:51
From: Sam
Subject: Long division to find power series

Can you show me how to find a power series to approximate this 
function:

   1/(1-x)^2

using long division.

I can do it for functions such as 1/(1+x):
      
        _1 - x + x^2 - x^3 + . . .__
   1+x ) 1
         1 + x
           --------
           - x
           - x - x^2
               -----------
                 x^2
                 x^2 + x^3
                     ---------
                     - x^3
                     - x^3 - x^4 . . . 

Can you show me how to do that other function the above way?

The power series for that function is 1 + 2x + 3x^2 + 4x^3 + .... to 
infinity.

Thanks,
Sam

Date: 09/01/98 at 02:18:05
From: Doctor Floor
Subject: Re: Long division to find power series

Thank you for sending in your question.

It is a little bit weird to do an infinite long division. When we do 
that, then we use (x-1)^2 = 1 - 2x + x^2:

               __1 + 2x + 3x^2 + 4x^3 + 5x^4 + ....____
  1 - 2x + x^2 ) 1
                 1 - 2x + x^2
                 ------------ -
                     2x - x^2
                     2x - 4x^2 + 2x^3
                     ---------------- -
                          3x^2 - 2x^3
                          3x^2 - 6x^3 + 3x^4
                          ------------------ -
                                 4x^3 - 3x^4
                                 4x^3 - 8x^4 + 4x^5
                                 ------------------ -
                                        5x^4 - 4x^5
                                        5x^4 - 10x^5 + 5x^6
                                        ------------------- -
                                               6x^5  - 5x^6
                                               ......

I hope this helps. If you have a math question again, please send it to 
Dr. Math.

Best regards,

- Doctor Floor, The Math Forum      
Check out our web site! http://mathforum.org/dr.math/