28 9. (a) Given that | z | < 1 , write down the sum of the infinite series 1 + z + z2 + z3 + … (1) (b) Given that z = 1 2 (cos θ + i sin θ) , (i) use the answer to part (a), and de Moivre’s theorem or otherwise, to prove that 1 2 sin θ + 1 4 sin 2θ + 1 8 sin 3θ + …    = 2sin 5 4cos θ θ − (5) (ii) show that the sum of the infinite series 1 + z + z2 + z3 + … cannot be purely imaginary, giving a reason for your answer. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________