Complex Power Series

Question

So, I'm trying to find the power series of ${1\over 1-z+z^2} around the point z=0.$ After some rather easy algebra I've determined the expression to be $${1\over z-(1+i\sqrt{3})/2} {1\over z-(1-i\sqrt{3})/2}$$ So I tried to use the ${1\over 1-x}$ power series as basis. Thing is, the result that I obtain is completely different from the one the solutions provide, so if anyone could clarify the oprocedure I would be extremely grateful.

Answer 1

HINT:

Assuming $|-z|<1,$

$$\frac1{1-z+z^2}=\frac{1+z}{1+z^3}=(1+z)(1+z^3)^{-1}$$

Answer 2

Hint: You can use partial fractions then use the geometric series.