# Taylor series representation

The Subject

## Homework Statement

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

## Homework Equations

$$\sum c_n (x-a)^n$$

## The Attempt at a Solution

$$\frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2}$$
since $$\frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2}$$
$$x*\frac{d}{dx}\frac{1}{(1+4x)^2} =x\frac{d}{dx}\sum_{n=0}^\infty(-4)^nx^n=x\sum_{n=0}^\infty(-4)^nnx^{n-1}=\sum_{n=0}^\infty(-4)^nnx^{n}$$

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

Am i doing something incorrect?