R is the reals. f :(0, 1) → R is differentiable and has a bounded derivative: |f '(x)| <= k. Prove that : |∫01 f(x) dx - ∑1n f(i/n) /n| ≤ k/n.
The worst case for the difference between 1/n f(i/n) and ∫i/n-1/ni/n f(x) dx is if f '(x) = k (or -k) for the entire range, in which case the difference is the area of a triangle base 1/n and height k/n. Hence the difference for the complete Riemann sum is at worst k/(2n).
Comment. Note that the question gives a result which is needlessly too poor by a factor 2.
7th Putnam 1947
© John Scholes
5 Mar 2002