8. Let α ∈ (−1, 1). Derive the real series ∞ 2 \ αn cos nθ = α cos θ − α 1 − 2α cos θ + α2 n=1 for all θ ∈ R. (Hint: Consider a Laurent series for α/(z − α) for |z| > |α|. Substitute z = eiθ .)