The Euler-Maclaurin Formula and Sums of Powers Revisited
نویسندگان
چکیده
Using the Euler-Maclaurin summation formula the strictly increasing convergence lim m → ∞ m j=1 j m m = e e − 1 is demonstrated.
منابع مشابه
Asymptotic Euler-Maclaurin formula for Delzant polytopes
Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler-Maclaurin formulas. An asymptotic Euler-Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin-Sternberg [GS]. Then, the problem is to find a concrete formula for the each term of the e...
متن کاملOn Reciprocity Formulas for Apostol’s Dedekind Sums and Their Analogues
Using the Euler-MacLaurin summation formula, we give alternative proofs for the reciprocity formulas of Apostol’s Dedekind sums and generalized Hardy-Berndt sums s3,p(b, c) and s4,p(b, c). We also obtain an integral representation for each sum.
متن کاملSum-integral Interpolators and the Euler-maclaurin Formula for Polytopes
A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space V , namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in V . The paper introduces the notion of an interpolator between...
متن کاملNotes on Euler-boole Summation
We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases in a general approach to approximation that also encompasses Taylor sums. Here we give additional details of the construction.
متن کاملEuler-Boole Summation Revisited
We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases of a general approach to approximation. Herein we give details and extensions of this idea.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010