- (a) Given that \(| z | < 1\), write down the sum of the infinite series
$$1 + z + z ^ { 2 } + z ^ { 3 } + \ldots$$
(b) Given that \(z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )\),
- use the answer to part (a), and de Moivre's theorem or otherwise, to prove that
$$\frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$
- show that the sum of the infinite series \(1 + z + z ^ { 2 } + z ^ { 3 } + \ldots\) cannot be purely imaginary, giving a reason for your answer.