Given a rational function p(x)q(x), what would the xk coefficient of this rational function mean (in particular for the negative k's). Is there some sort of expansionp(x)q(x)=∑−∞∞akxkwhere one would say ak is the coefficient of xk?How would you find the coefficient? A very simple rational function would be (x−1)(x−2)(x−3). How would you find the x−1 coefficient of this?Need help!

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Given a rational function p(x)q(x), what would the xk coefficient of this rational function mean (in particular for the negative k&#039;s). Is there some sort of expansionp(x)q(x)=∑−∞∞akxkwhere one would say ak is the coefficient of xk?How would you find the coefficient? A very simple rational function would be (x−1)(x−2)(x−3). How would you find the x−1 coefficient of this?Need help!

emeriinb4r · Accepted Answer

There is no such expansion that converges everywhere. To see this, consider what happens to both sides of your equation as x→0 and x→∞.    You can, however, expand p(x)q(x) as different series that converge around different values of x. For instance, near 0 you have    (x−1)(x−2)(x−3)=∑k=0∞[12(12)k−23(13)k]xk,    which you can derive by using partial fractions and the formula for geometric series. Similarly expanding p(y−1)q(y−1) with respect to y will give you a power series in x−1 valid as x→∞:    p(y−1)q(y−1)=−16+2311−3y−1211−2y=−16+∑k=0∞[233k−122k]yk=−16+∑k=0∞[233k−122k]x−k    So in some sense you could say that the coefficient of x−1 is 1, near infinity.

Jonas Burt · Accepted Answer

Every rational function has an expansion of the form  ∑k=r∞akxk where r may be positive, zero, or negative, but is in any case finite. If r≥0 then the coefficient of x−1 is zero. If r&amp;lt;0 then a−1 can be found by the following procedure: let s=−r, multiply the rational function by xs, differentiate s−1 times, evaluate at x=0, and divide by s−1 factorial.

Given a rational function \frac{p(x)}{q(x)}, what would the x^k coefficient of this rati

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