Maclaurin series representation

Question

I'm trying to find Maclaurin series for function $f(x)=\frac{1}{x^2+x+1}$.

I've got so far $$f(x)=\frac{1}{x^2+x+1}=\frac{x-1}{x^3-1}=\frac{x}{x^3-1}-\frac{1}{x^3-1}=-\frac{x}{1+(-x^3)}+\frac1{1+(-x^3)}=$$ $$=-\sum_{n=0}^{\infty}(-1)^n x^{3n+1}+\sum_{n=0}^{\infty}(-1)^n x^{3n}=\sum_{n=0}^{\infty}(-1)^n x^{3n}(-x+1)$$

Answer 1

\begin{align} f(x) &= \frac{1}{1+x+x^{2}} = \frac{1-x}{1-x^{3}} = (1-x) \sum_{n=0}^{\infty} x^{3n} \\ &= \sum_{n=0}^{\infty} (x^{3n} - x^{3n+1}). \end{align}