2023 TGM P1 Q8

Consider the complex series \({{{e}^{\mathrm{i}\theta }}+\frac{1}{2}{{e}^{\mathrm{2i}\theta }}+\frac{1}{4}{{e}^{\mathrm{3i}\theta }}+\frac{1}{8}{{e}^{\mathrm{4i}\theta }}+…}\)

1. Find \({{{u}_{r}}}\) in terms of \({r}\), where \({{{u}_{r}}}\) is the \({r}\)th term of the series.[1]
2. Show that the series is geometric and convergent. Hence, state the sum to infinity.[3]
3. By expressing the sum to infinity in terms of \({\sin \theta }\) and \({\cos \theta }\), show that
   \({\cos \theta +\frac{1}{2}\cos 2\theta +\frac{1}{4}\cos 3\theta +…=\frac{4\cos \theta -2}{5-4\cos \theta }}\).[5]