# Taylor series representation

Tags:
1. Aug 5, 2016

### The Subject

1. The problem statement, all variables and given/known data
Find a power series that represents $$\frac{x}{(1+4x)^2}$$

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

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

Am i doing something incorrect?

2. Aug 5, 2016

### Staff: Mentor

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