26th Putnam 1965 Problem B4 Define f n : R → R by f n (x) = (nC0 + nC2 x + nC4 x 2 + ... ) / (nC1 + nC3 x + nC5 x 2 + ... ), where nCm is the binomial coefficient. Find a formula for f n+1 (x) in terms of f n (x) and x, and determine lim n→∞ f n (x) for all real x.
It is almost obvious that f n+1 = (f n + x)/(f n + 1) (*). So if f n (x) tends to a limit k(x), then k(x) = ( k(x) + x) /( k(x) + 1), and hence k(x) = √x. Obviously, f n (0) = 1/n, so k(0) = 0. We notice also that f n (x) = (√x) N/D, where N = (1 + √x) n + (1 - √x) n and D = (1 + √x) n - (1 - √x) n . Suppose 0  1, put y = (√x - 1)/(1 + √x) and we again get k(x) = √x. It is clear from (*), that for x < 0, f n (x) does not tend to a limit. 26th Putnam 1965 © John Scholes jscholes@kalva.demon.co.uk 25 Jan 2002