Taylor Series for $\dfrac{x}{x ^ 2 + x + 1}$. I'm trying to calculate the taylor series of $\dfrac{x}{x ^ 2 + x + 1}$.
Algebraic manipulation didn't get me anywhere, since the roots of $ x ^ 2 + x + 1 $ are complex.
Integrate or derive made the problem worse
Any tips on how to proceed?

There seem to be many ways to go about this, so here is one: put $\dfrac{x}{x ^ 2 + x + 1}=\sum_{n=0}^{\infty} a_nx^n$, then $$x=\sum_{n=0}^{\infty} a_nx^n(x^2+x+1)=a_0+(a_1+a_0)x+\sum_{n=2}^{\infty}(a_n+a_{n-1}+a_{n-2})x^n,$$
and by comparing the coefficients we get $a_0=0$, $a_1=1$, and $a_n+a_{n-1}+a_{n-2}=0$ for $n \geq 2$. You can see the coefficients repeat ($a_2=-1,a_3=0,a_4=1,\dots$), so we have $a_{3k}=0, a_{3k+1}=1, a_{3k+2}=-1$, or in other words 
$$
\dfrac{x}{x ^ 2 + x + 1}=x-x^2+x^4-x^5+\dots
$$
