- 32

- 0

**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?