A geometric sequence has first term 1 and common ratio 0.75.
(i) Find the sum to infinity, S∞S_{\infty}S∞, of the sequence.
(ii) Hence, or otherwise, evaluate
∑n=1∞(cos30∘)2n \sum_{n=1}^{\infty} (\cos 30^{\circ})^{2n} n=1∑∞(cos30∘)2nFind the smallest positive exact value of θ\thetaθ, in radians, which satisfies the equation
∑n=0∞(2cosθ)n=22−2 \sum_{n=0}^{\infty} (\sqrt{2}\cos \theta)^n = \frac{2}{2 - \sqrt{2}} n=0∑∞(2cosθ)n=2−22
