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### 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/
```
Associated Topics:
High School Sequences, Series

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