Difficulty finding a power series representation for a function

enharmonics

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As the title says, I'm having a lot of trouble finding a power series representation for a function.

Here it is:

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And here's my attempt at a solution:

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I had a lot of trouble getting the hang of reindexing, but after studying up a bit, the above is what I ultimately got.

Is this correct? If so, how would I go about finding the interval of convergence? Would it just be with the ratio test as usual? It seems like it would be quite difficult to find it that way, since I've got so many terms in my final summation.
 

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5) Use the methods of section 11.9 to find, with excruciating details, a power series representation for the function:

. . . . .f(x)=xx2(1+2x)3\displaystyle f(x)\, =\, \dfrac{x\, -\, x^2}{(1\, +\, 2x)^3}

Include the interval of convergence.
What are "the methods of section 11.9"? Thank you! ;)
 
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