5 State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\). By letting \(z = \cos \theta + \mathrm { i } \sin \theta\), show that $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos n \theta = \frac { \sin \frac { 1 } { 2 } n \theta } { \sin \frac { 1 } { 2 } \theta } \cos \frac { 1 } { 2 } ( n + 1 ) \theta$$ where \(\sin \frac { 1 } { 2 } \theta \neq 0\).