48th Putnam 1987

Define the sequences x_i and y_i as follows. Let (x₁, y₁) = (0.8, 0.6) and let (x_n+1, y_n+1) = (x_ncos y_n - y_nsin y_n, x_nsin y_n + y_n cos y_n) for n ≥ 1. Find lim_n→∞ x_n and lim_n→∞ y_n.

Solution

Put y₀ = sin^-10.6. Then the relations give immediately x_n+1 = cos(y₀ + y₁ + ... + y_n), y_n+1 = sin(y₀ + y₁ + ... + y_n).

Since sin x < x for x > 0, we have sin x < π - x for 0 < x < π. Hence y₀ + y₁ + ... + y_n < π implies y₀ + y₁ + ... + y_n+1 < π. So all y_n are positive. Hence ∑ y_n is monotone increasing and bounded above. So it converges. Hence y_n → 0. But that implies that ∑ y_n → π, and hence x_n → -1.

48th Putnam 1987

© John Scholes
jscholes@kalva.demon.co.uk
7 Jan 2001