A note on the Newton-Cotes integration formula (Q1176338)

This paper presents a simple example of a continuous function \(f\) on [- 1,1] such that the Newton-Cotes integration formula does not converge. This example is based on the well-known result of C. Runge related to the uniform divergence of the Lagrangian interpolating polynomials of the function \(h(x)=(1+25x^ 2)\), \(-1\leq x\leq 1\), for equally spaced nodes. In fact, a necessary and sufficient condition is obtained for the convergence of the Newton-Cotes formula on functions of the form \(f(x)=(1+N^ 2x^ 2)^{-1}\), \(N>0\).