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

Asked Mar 08 '16 at 20:21

Active Mar 08 '16 at 20:21

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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!

asked Mar 08 '16 at 20:21

Lauren Hayes

4

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 08 '16 at 20:25
2

Thank you most kindly, I should be able to get it from here! – Lauren Hayes Mar 08 '16 at 20:26
4

Perfect. (Your comment is something one reads all too rarely on the site. As such, it is most refreshing...) – Did Mar 08 '16 at 20:28
Alternatively, do a partial fractions decomposition. – Mar 09 '16 at 03:14
Possible duplicate of Find the power series of $f(x)=\frac{1}{x^2+x+1}$ – May 20 '18 at 09:22

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

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