Expanding Power Series for (x+x^2)/(1-x)^3 | Simplification Techniques

[clarinetqueen]

Homework Statement

Expand f(x)= (x+x²)/(1-x)³

Homework Equations

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The Attempt at a Solution

I've tried everything I can think of to simplify this equation: substitution of various other power series, partial fraction decomposition, taking derivatives, multiplying out the denominator. It's driving me nuts. Thanks for any steps in a new direction.

 

[micromass]

OK, let's do this in steps. First, can you find the power series expansion of $\frac{1}{(1-x)^3}$??

If you can't, try to use that

$$\frac{1}{1-x}=1+x+x^2+x^3+...$$

and differentiate both sides.

 

[clarinetqueen]

I tried to find a power expansion for that term and found it as 2(1-x)^3, which would equal the power series (n^(2)-n)x^(n-2) from n=2. From here, I'm not sure if there is an equation for the numerator, or if there is a way to multiply the numerator into this equation in a way that makes sense.

 

[micromass]

OK, and now you simply can do

$$\frac{x+x^2}{(1-x)^3}=x\frac{1}{(1-x)^3}+x^2\frac{1}{(1-x)^3}$$

So exchange both terms of $\frac{1}{(1-x)^3}$ by its power series and multiply and add everything.

 

[clarinetqueen]

Oh wow. Thanks for your help, I guess my brain is rebelling against obvious steps.

Ok, so now I have the sum of (n^2-n)x^(n-1) + sum of (n^2-n)x^n, both starting from n=2, and both expressions multiplied by 1/2.

 

[clarinetqueen]

Ok, so I have the sums 1/2(n^(2)-n)x^(n-1) + 1/2(n^(2)-n)x^(n-1) from n=2. The next part of the question says I should relate (n^2)/(2^n) to the previous equation. Is there a simplification I'm missing?

 

[micromass]

Ok, so I have the sums 1/2(n^(2)-n)x^(n-1) + 1/2(n^(2)-n)x^(n-1) from n=2. The next part of the question says I should relate (n^2)/(2^n) to the previous equation. Is there a simplification I'm missing?

Well, you will have to put x=2. Try to rewrite the equation a bit to see if you get anything nice.