42nd Putnam 1981 Problem A3 Evaluate: lim k→∞ e -k ∫ R (e x - e y ) / (x - y) dx dy, where R is the rectangle 0 ≤ x, y, ≤ k. Solution
: It diverges to plus infinity. We use L'Hôpital's rule. Let f(k) = ∫ R (e x - e y ) / (x - y) dx dy, then the limit is the same as the limit of f'(k)/e k . To work out f'(k), note that f(k + δk) - f(k) = ∫ δR (e x - e y ) / (x - y) dx dy, where δR is the strip from x = 0 to x = k at y = k, width δk and the strip from y = 0 to k at x = k width δk. The integral over the two strips is obviously the same, so we get f'(k) = 2 ∫ 0 k (e k - e x ) / (k - x) dx. Hence f'(k)/e k = 2 ∫ 0 k (1 - e -(k-x) ) / (k - x) dx. Changing the integration variable to z = k - x, we get 2 ∫ 0 k (1 - e -z ) / z dz. The integrand is always positive and for z >= 1 is at least 1 - 1/e > 1/2. So the integral is at least ∫ 1 k 1/z dz = ln k. This tends to infinity with k. 42nd Putnam 1981 © John Scholes jscholes@kalva.demon.co.uk 16 Jan 2001