Power Series from Long DivisionDate: 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/ |
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