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## Homework Statement

Find the coefficient of x^n in the expansion of each of the following functions as a series of ascending powers of x.

[itex]\frac{1}{(1+2x)(3-x)}[/itex]

## Homework Equations

## The Attempt at a Solution

[itex](1+2x)^{-1} = 1 + (-1)2x + \frac{(-1)(-2)}{2!}(2x)^2 + \frac{(-1)(-2)(-3)}{3!}(2x)^3... -1<2x<1[/itex]

[itex]= 1 - 2x + 4x^2 - 8x^3... -\frac{1}{2}<x<\frac{1}{2}[/itex]

[itex]\frac{1}{3}(1-\frac{x}{3})^{-1} = \frac{1}{3}(1 + (-1)(-\frac{x}{3}) + \frac{(-1)(-2)}{2!}(-\frac{x}{3})^2 + \frac{(-1)(-2)(-3)}{3!}(-\frac{x}{3})^3...) -1<-\frac{x}{3}<1[/itex]

[itex]= \frac{1}{3} + \frac{x}{9} + \frac{x^2}{27} + \frac{x^3}{81}... -3<x<3[/itex]

[itex]\frac{1}{(1+2x)(3-x)}=\frac{1}{3}-\frac{5}{9}x+\frac{31}{27}x^2-\frac{185}{81}x^3... -\frac{1}{2}<x<\frac{1}{2}[/itex]

I can't see an obvious pattern in the numerators of the coefficients, perhaps I should be able to. So I look at the coefficient of each binomial where I see that the coefficient of [itex]x^n[/itex] in the expansion of [itex](1+2x)^{-1}[/itex] is [itex](-2)^n[/itex] and the coefficient of [itex]x^n[/itex] in the expansion of [itex](3-x)^{-1}[/itex] is [itex](\frac{1}{3})^{n+1}[/itex]. So [itex]\frac{1}{(1+2x)(3-x)}[/itex] could also be expressed as [itex](\sum^{∞}_{r=0}(-2x)^r)(\sum^{∞}_{r=0}(\frac{1}{3})^{r+1}(x)^r) -\frac{1}{2}<x<\frac{1}{2}[/itex]. However as I feel I am getting tantalisingly close to a solution I realise that their multiplication complicates matters.

So I start to look at patterns in the process of multiplying the 2 expansions together and find that

[itex]\frac{1}{(1+2x)(3-x)}=\frac{1}{3}(\sum^{∞}_{r=0} (-1)^r(2x)^r(\sum^{∞}_{k=0} (\frac{x}{3})^k)) -\frac{1}{2}<x<\frac{1}{2}[/itex]

By now I've strayed into territory outside of my text book, ie the recursive summation, this might well be meaningless as it looks like it never bottoms out of the k loop, but I hope it illustrates my thought process, however Ill conceived it might be.

I thought there might be a way to eliminate the k in the last expression to yield my general coefficient.

Any guidance on the approach to tackling this kind of problem would be much appreciated.