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Taylor series representation

1. Homework Statement
Find a power series that represents $$ \frac{x}{(1+4x)^2}$$

2. Homework Equations
$$ \sum c_n (x-a)^n $$

3. The Attempt at a Solution
$$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$
since [tex] \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} [/tex]
$$ x*\frac{d}{dx}\frac{1}{(1+4x)^2} =x\frac{d}{dx}\sum_{n=0}^\infty(-4)^nx^n=x\sum_{n=0}^\infty(-4)^nnx^{n-1}=\sum_{n=0}^\infty(-4)^nnx^{n}$$

The solution suggests $$\sum_{n=0}^\infty(-4)^n(n+1)x^{n+1}$$

Am i doing something incorrect?
 

fresh_42

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Reconsider your differentiation. Isn't ##\frac{d}{dx} x^{-2} = -2x^{-3}##?
 

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