Multiplication of Laurent Series

[libelec]

Homework Statement

Find the expression of the Laurent series for f(z) = \frac{z}{{(2z - 1)(\frac{2}{z} - 1)}} so that \sum\limits_{ - \infty }^\infty {{a_n}} \ converges.

The Attempt at a Solution

First, I find that z = 1/2 and z = 2 (and infinite) are poles of the function f. Then, I can rewrite f(z) = \frac{1}{{(2z - 1)(2 - z)}} since it doesn't change anything in terms of poles (is this right?). After that, I calculate the first Laurent series for \frac{1}{{(2z - 1)}} and for \frac{1}{{(2 - z)}} (the one for |z| < 1/2) and I find that f(z) = - \frac{1}{2}\sum\limits_{n = 0}^\infty {{2^n}{z^n}} \sum\limits_{n = 0}^\infty {\frac{{{z^n}}}{{{2^n}}}}. Using the Cauchy product rule, I find that a_n has an expression for even powers and another for odd powers. Therefore, it diverges (right?)

I look for the second Laurent series (1/2 < |z| < 2). The expression of \frac{1}{{(2 - z)}} is the same, but I have to change the other one. I find that f(z) = \frac{1}{{4z}}\sum\limits_{n = 0}^\infty {\frac{1}{{{2^n}{z^n}}}} \sum\limits_{n = 0}^\infty {\frac{{{z^n}}}{{{2^n}}}}.

Now, one of the series is all of negative powers and the other one of positive powers. Even more, they have the same powers in ||. My question is, can I just simplify the powers of z and the powers of 2? Or can I use the Cauchy product rule?

Thanks

 

[libelec]

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[libelec]

Does anybody have any ideas?