Problem with the Laurent series

Question

Find The Laurent series for the following function on the annulus $1<|z|<2$ :

$\displaystyle f(z)=\frac{2z}{z^2+z-2}$

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My work :

$\displaystyle f(z)=\frac{2}{3} \left( \frac{1}{z-1}+\frac{2}{z+2}\right)=\frac{2}{3} \left(\sum _{k=-\infty }^{-1} z^k+\sum _{k=0}^{\infty } \left(-\frac{z}{2}\right)^k\right) \ \ \ \ \ \ \ (\text{Geometric series})$

Ratio test can prove that it converges.

Please check my work before posting the solution.

Thank you.

Answer 1

The decomposition is ok and the development into Laurant series is ok too.