Convergence rates of approximate sums of Riemann integrals

The Riemann sums and the trapezoidal sums of functions defined on a bounded closed interval are well known as approximate sums of the Riemann integrals of the functions. In this paper the author represents the convergence rates of the Riemann sums and the trapezoidal sums as some limits of their expanded error terms. Let [a, b] be a bounded closed interval. We take an n-division ∆ of [a, b] defined by ∆ : a = s0 ≤ s1 ≤ · · · ≤ sn−1 ≤ sn = b. We denote by Dn the division of [a, b] defined by si = a+ i(b− a)/n and call it the regular n-division. For a function f defined on [a, b] and si−1 ≤ ξi ≤ si we define the Riemann sum R(f ; ∆, ξi) by