Euler-Maclaurin summation formulaEulerMaclaurinSummationFormula2001-10-15 20:48:412002-05-25 20:51:12Theoremnumber theory\usepackage{amssymb}
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\usepackage{xypic}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
\[
\sum_{a < n \leq b} f(n) = \int_a^b f(t)dt + \sum_{r=0}^k \frac{(-1)^{r+1}B_{r+1}}{(r+1)!}(f^{(r)}(b) - f^{(r)}(a)) + \frac{(-1)^k}{(k+1)!} \int_a^b B_{k+1}(t)f^{(k+1)}(t)dt. \]