4th Putnam 1941 Problem A2 Define f(x) = ∫ 0 x ∑ i=0 n-1 (x - t) i / i! dt. Find the nth derivative f (n) (x).
Straightforward. Note that x appears both in the integrand and in the limits, so a little care is needed. Write g r (x, t) = ∑ i=0 r-1 (x - t) i / i! so that f(x) = ∫ 0 x g n (x, t) e nt dt. By definition f '(x) = lim δx -> 0 (∫ 0 x+δx g n (x+δx, t) e nt dt - ∫ 0 x g n (x, t) e nt dt) / δx = e nx g n (x, x) + ∫ 0 x g n '(x,t) e nt dt, where the ' denotes the partial derivative wrt the first variable. But g n (x, x) = 1, and g r '(x, t) = g r-1 (x, t), so f '(x) = e nx + ∫ 0 x g n-1 (x, t) e nt dt. Hence by an easy induction f (n) (x) = e nx ( 1 + n + n 2 + ... + n n-1 ). 4th Putnam 1941 © John Scholes jscholes@kalva.demon.co.uk 15 Sep 1999