Asymptotics of Riemann Sums for Uniform Partitions of Intervals of Integration

Our goal is to prove the Euler–Maclaurin summation formula using only the Taylor formula. Furthermore, the Euler–Maclaurin summation formula will be considered for the case of functions of the class C2k([a,b]). A stronger version of the estimation of the rest for functions of class Cn([a,b]) is given. We will also present the application of the obtained Euler–Maclaurin summation formulas to determine the asymptotics of Riemann sums for uniform partitions of intervals of integration. This paper also introduces the concepts of the asymptotic smoothing of the graph of a given function, as well as the asymptotic uniform distribution of the positive and negative values of the given function (generalizing the concept of the symmetric distribution of these values with respect to the x-axis, as in the case of the Peano–Jordan measure).