Sum geometric series with complex terms

The infinite series \(C\) and \(S\) are defined by $$C = \cos \theta + \frac{1}{2}\cos 5\theta + \frac{1}{4}\cos 9\theta + \frac{1}{8}\cos 13\theta + \ldots$$ $$S = \sin \theta + \frac{1}{2}\sin 5\theta + \frac{1}{4}\sin 9\theta + \frac{1}{8}\sin 13\theta + \ldots$$ Given that the series \(C\) and \(S\) are both convergent,

1. show that $$C + iS = \frac{2e^{i\theta}}{2 - e^{4i\theta}}$$ [4]
2. Hence show that $$S = \frac{4\sin \theta + 2\sin 3\theta}{5 - 4\cos 4\theta}$$ [4]

In this question you must show detailed reasoning.

1. Show that \(e^{i\theta} - e^{-i\theta} = 2i\sin\theta\). [1]
2. Hence, show that \(\frac{2}{e^{2i\theta} - 1} = -(1 + i\cot\theta)\). [3]
3. Two series, \(C\) and \(S\), are defined as follows. $$C = 2 + 2\cos\frac{\pi}{10} + 2\cos\frac{\pi}{5} + 2\cos\frac{3\pi}{10} + 2\cos\frac{2\pi}{5}$$ $$S = 2\sin\frac{\pi}{10} + 2\sin\frac{\pi}{5} + 2\sin\frac{3\pi}{10} + 2\sin\frac{2\pi}{5}$$ By considering \(C + iS\), find a simplified expression for \(C\) in terms of only integers and \(\cot\frac{\pi}{10}\). [8]
4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. [2]

Given that \(w_n = 3^{-n} \cos 2n\theta\) for \(n = 1, 2, 3, \ldots\), use de Moivre's theorem to show that $$1 + w_1 + w_2 + w_3 + \ldots + w_{N-1} = \frac{9 - 3\cos 2\theta + 3^{-N+1} \cos 2(N-1)\theta - 3^{-N+2} \cos 2N\theta}{10 - 6\cos 2\theta}.$$ [7] Hence show that the infinite series $$1 + w_1 + w_2 + w_3 + \ldots$$ is convergent for all values of \(\theta\), and find the sum to infinity. [2]