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$$
=-\sum _{{m=1}}^{{\infty }}\left ({\frac  {z}{0.5}}\right )^{{m}}=-{\frac  {0.5^{{-1}}z}{1-0.5^{{-1}}z}}={\frac  {z}{z-0.5}}={\frac  {1}{1-0.5z^{{-1}}}}\ 
$$
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