Asymptotic Behaviors of Intermediate Points in the Remainder of the Euler-Maclaurin Formula

The Euler-Maclaurin formula is a very useful tool in calculus and numerical analysis. This paper is devoted to asymptotic expansion of the intermediate points in the remainder of the generalized Euler-Maclaurin formula when the length of the integral interval tends to be zero. In the special case we also obtain asymptotic behavior of the intermediate point in the remainder of the composite trapezoidal rule. 1. Introduction It is well known that the Euler-Maclaurin formula is a formula used in the numerical evaluation of integral, which states that the value of an integral is equal to the sum of the value given by the trapezoidal rule and a series of terms involving the odd-numbered derivatives of the function at the end points of the integral interval. Specifically, for the function the Euler-Maclaurin formula can be expressed as follows where and is some point between and . The constants are known as Bernoulli numbers, which are defined by the equation The first few of the Bernoulli numbers are , , , and for all . The Euler-Maclaurin formula was discovered independently by Leonhard Euler and Colin Maclaurin, and it has wide applications in calculus and numerical analysis. For example, the Euler-Maclaurin formula is often used to evaluate finite sums and infinite series when and are integers. Conversely, it is also used to approximate integrals by finite sums. Therefore, the Euler-Maclaurin formula provides the correspondence between sums and integrals. Besides, the Euler-Maclaurin formula may be used to derive a wide range of quadrature formulas including the Newton-Cotes formulas, and used for detailed error analysis in numerical quadrature. The Euler-Maclaurin formula has many generalizations and extensions [1–6]. A direct generalization of the Euler-Maclaurin formula in the interval can be described as where , is a positive integer and is a nonnegative integer. Obviously, when , (1.4) reduces to (1.1). We also conclude that this equation has algebraic accuracy of which is the same as (1.1). Recently, some interests have been focused on the study of the mean value theorem for integrals and differentiations [7–14]. The aim of the present paper is to deal with asymptotic expansions of the intermediate points in the generalized Euler-Maclaurin formula when the length of the integral interval tends to be zero. The rest of this paper is organized as follows. In the second section, the Bell polynomials as a standard mathematical are introduced in detail. In the third section, we give asymptotic behavior of the intermediate points in the remainder of the