46th Putnam 1985 Problem A3 x is a real. Define a i 0 = x/2 i , a i j+1 = a i j 2 + 2 a i j . What is lim n→∞ a n n ? Solution
: e x - 1. a n+1 n for x is the same as a n n for x/2. So if we write a n n as p n (x) then evidently p n is a polynomial and we have the relation p n+1 (x) = 2p n (x/2) + p n (x/2) 2 (*) and also p 0 (x) = x. If we add 1 to both sides of (*), then it becomes 1 + p n+1 (x) = (1 + p n (x/2) ) 2 . Iterating: 1 + p n (x) = (1 + x/N) N , where N = 2 n . But lim (1 + x/n) n = e x as n → ∞, so p n (x) → e x - 1. 46th Putnam 1985 © John Scholes jscholes@kalva.demon.co.uk 7 Jan 2001