# Writing $\frac{1}{1 + w + w^2}$ as a power series and finding the ROC

I have to write the following:

$\frac{1}{1 + w + w^2}$

as a power series: $$\sum_{n=0}^{\infty}{a_nw^n}$$

and find the radius of convergence of the series (in the complex plane). Obviously you can use the geometric series formula to obtain the following series:

$$\sum_{n=0}^{\infty}(-1)^n(w + w^2)^n$$

which converges iff $|w + w^2| \lt 1$. However, I can't figure out a way to write it in the power series form above and also I cannot find the ROC in terms of an inequality just involving $w$. Is this possible to do? Thanks for your help!

• More promising (and actually leading to the result):$$\frac{1}{1 + w + w^2}=(1-w)\cdot\frac1{1-w^3}=(1-w)\cdot\sum_{n=0}^\infty\ldots$$ – Did Mar 8 '16 at 20:25
• Thank you most kindly, I should be able to get it from here! – Lauren Hayes Mar 8 '16 at 20:26
• Perfect. (Your comment is something one reads all too rarely on the site. As such, it is most refreshing...) – Did Mar 8 '16 at 20:28
• Alternatively, do a partial fractions decomposition. – user138530 Mar 9 '16 at 3:14
• Possible duplicate of Find the power series of $f(x)=\frac{1}{x^2+x+1}$ – user99914 May 20 '18 at 9:22