Challenging +1.3 This is a standard Further Maths question on geometric series with complex numbers using de Moivre's theorem. While it requires multiple steps (expressing cosines using complex exponentials, summing a geometric series, and simplifying), the technique is well-established and follows a predictable pattern. The convergence part is straightforward once the first part is complete. More routine than the average Further Maths proof question but still requires solid technique.
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]
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]
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q9 [9]}}