Let 2f(n) be the highest power of 2 dividing n. Let g(n) = f(1) + f(2) + ... + f(n). Prove that ∑ exp( -g(n) ) converges.
If n is odd, then f(n) = 0. If n is even, then f(n) ≥ 1. Hence g(n+2) ≥ g(n) + 1. So the series is dominated by 1 + (1/e + 1/e) + (1/e2 + 1/e2) + ... < 2/(1 - 1/e). So the sums are bounded above. All terms are positive, so it must converge.
21st Putnam 1960
© John Scholes
15 Feb 2002