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I don't understand what happens during series multiplication and substitution. It doesn't seem to make sense. Shouldn't it just be $\large(x^{n})^{n}$?

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See this: en.wikipedia.org/wiki/Cauchy_product –  Cortizol Feb 11 '13 at 19:51
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Please post questions as text and LaTeX instead of just a big image. As it is, the question is unsearchable and a strain to read. –  Antonio Vargas Feb 11 '13 at 20:35

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The comment by Cortizol explains how to multiply two power series together. However, the solution above is dull as dishwater. Notice that $\frac{1}{(1-x)^2} = \frac{d}{dx}\frac{1}{1-x}$. Thus,

$$\frac{1}{(1-x)^2}=\frac{d}{dx}\left(\frac{1}{1-x}\right)=\frac{d}{dx}(1+x+x^2+x^3+x^4+\ldots)=1+2x+3x^2+4x^3+\ldots$$

which can be written as $\sum_{i=0}^\infty (i+1)x^{i}$.

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I think Im missing something here. Why are you finding the derivative? –  user1730308 Feb 11 '13 at 21:14
    
@user1730308, differentiating is just a way of reducing the problem to an expression whose power series is already known. –  Peter Phipps Feb 11 '13 at 21:36

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