The Euler–Maclaurin Formula of Arbitrary Order

Abstract

In this chapter we extend the results formulated in Chap. 12 to order m, i.e., to functions of class $\mathscr {C}^m$, for any $m\in \mathbb N_{\ge 1}$. Namely, we formulate the Euler–Maclaurin formula, its asymptotic version, its applications to the approximation of finite sums and of sums of convergent series in great generality. The reader will find here not only the general statements of the formulas of order m, but also their detailed proofs, some of which were just sketched out or even skipped entirely in the previous sections. Among the applications, we discover the Hermite formula for the approximations of integrals: it is a refinement of the trapezoidal method which is even more accurate than Simpson’s method.