Euler-Maclaurin summation formula

Let $B_{r}$ be the $r\mbox{th}$ Bernoulli number, and $B_{r}(x)$ be the $r\mbox{th}$ Bernoulli periodic function. For any integer $k\geq 0$ and for any function $f$ of class $C^{k+1}$ on $[a,b],a,b\in\mathbb{Z}$ , we have

$\sum_{a<n\leq b}f(n)=\int_{a}^{b}f(t)dt+\sum_{r=0}^{k}\frac{(-1)^{r+1}B_{r+1}}% {(r+1)!}(f^{(r)}(b)-f^{(r)}(a))+\frac{(-1)^{k}}{(k+1)!}\int_{a}^{b}B_{k+1}(t)f% ^{(k+1)}(t)dt.$

Title	Euler-Maclaurin summation formula
Canonical name	EulerMaclaurinSummationFormula
Date of creation	2013-03-22 11:46:01
Last modified on	2013-03-22 11:46:01
Owner	KimJ (5)
Last modified by	KimJ (5)
Numerical id	9
Author	KimJ (5)
Entry type	Theorem
Classification	msc 65B15
Classification	msc 00-02
Related topic	BernoulliNumber