The Euler–Maclaurin Formulas of Order 1 and 2

Abstract

Let $a<b$ be two integers and $f:[a, b]\rightarrow \mathbb R$ a function. In Chap. 6 we saw how to calculate the sum $\displaystyle \displaystyle \sum _{a\le k<b}f(k)$ through the notion of a discrete primitive. In this chapter we instead study the problem of estimating such a sum. It is well known that when f is a monotonic function one can make such an approximation by way of the integral $\displaystyle \int _a^bf(t)\,dt$: in this case the modulus of the difference between the sum and the integral of f is at most equal to $|f(b)-f(a)|$. The integral formula may be extended for non-monotonic functions provided that they are regular, and is referred to as the Euler–Maclaurin formula. The Euler–Maclaurin formula represents both a method for calculating the sum $\displaystyle \sum _{a\le k<b}f(k)$ by means of the integral of f, and also a way of calculating the integral $\displaystyle \int _a^bf(x)\,dx$ once one knows the sum $\displaystyle \sum _{a\le k<b}f(k)$. These two different points of view gave Euler and Maclaurin their respective motivations for establishing the formula bearing their names, which was subsequently treated in detail [28] by Poisson. In this chapter we deal only with the Euler–Maclaurin formulas of order 1 and 2 for functions that are, respectively, of class $\mathscr {C}^1$ or $\mathscr {C}^2$: these cases are the source of most of the subsequent applications in the book. We treat the Euler–Maclaurin formula in full generality for functions of class $\mathscr {C}^m$ in the next chapter. In the last section we recover Euler–Maclaurin type formulas assuming just the monotonicity (no smoothness) of the function: the methods employed there are very different from those concerning regular functions and arise from the proof of the celebrated integral test for the convergence of series with monotonic terms. Last but not least, in many cases we make use of a CAS to perform hard computations (like difficult integrals, sums, sums of series). Some remarks about the notation used throughout the following chapters: remainders in formulas will be typically denoted by a letter R; we will replace the R by $\varepsilon$ if the remainder may tend to 0 by passing to the limit in some parameter, i.e., when the formula truly provides an approximation.

The Euler–MacLaurin summation formula is one of the most remarkable formulas of mathematics.

G.C. Rota [32]

Euler’s summation formula and its relation to Bernoulli numbers and polynomials provides a treasure trove of interesting enrichment material suitable for elementary calculus courses.

T.M. Apostol [3]