Find the Laurent expansion of \({\rm{f}}\left( {\rm{z}} \right) = \frac{{7{\rm{z}} - 2}}{{\left( {{\rm{z}} + 1} \right){\rm{z}}\left( {{\rm{z}} - 2} \right)}}\) in the region 1 < z + 1 < 3
1
\([\frac{{ - 2}}{{\left( {z + 1} \right)}} + \frac{1}{{{{\left( {z + 1} \right)}^2}}} + \frac{1}{{{{\left( {z + 1} \right)}^3}}} + \ldots \infty] - \frac{4}{3}\left[ {1 + \frac{{z + 1}}{3} + \frac{{{{\left( {z + 1} \right)}^2}}}{{{3^2}}} + \frac{{{{\left( {z + 1} \right)}^3}}}{3} + \ldots \infty } \right]\)
2
\( - \frac{4}{3}\left[ {1 + \frac{1}{{3\;\left( {z + 1} \right)}} + \frac{1}{{{3^2}{{\left( {z + 1} \right)}^2}}} + \frac{1}{{{3^3}{{\left( {z + 1} \right)}^3}}} + \ldots \infty } \right] + \left[ { - 2\left( {z + 1} \right) + {{\left( {z + 1} \right)}^2} + {{\left( {z + 1} \right)}^3} + \ldots \infty } \right]\)
3
\( - \frac{2}{3}\left[ {1 + \frac{1}{{3\;\left( {z + 1} \right)}} + \frac{1}{{{3^2}{{\left( {z + 1} \right)}^2}}} + \frac{1}{{{3^3}{{\left( {z+ 1} \right)}^3}}} + \ldots \infty } \right] + \left[ { - 2\left( {z + 1} \right) + {{\left( {z + 1} \right)}^2} + {{\left( {z + 1} \right)}^3} + \ldots \infty } \right]\)
4
\([\frac{{ - 2}}{{\left( {z + 1} \right)}} + \frac{1}{{{{\left( {z + 1} \right)}^2}}} + \frac{1}{{{{\left( {z + 1} \right)}^3}}} + \ldots \infty] - \frac{2}{3}\left[ {1 + \frac{{z + 1}}{3} + \frac{{{{\left( {z+ 1} \right)}^2}}}{{{3^2}}} + \frac{{{{\left( {z + 1} \right)}^3}}}{3} + \ldots \infty } \right]\)