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Euler-Maclaurin summation formula
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(Theorem)
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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$$ \sum_{a < n \leq b}
f(n) = \int_a^b f(t)dt + \sum_{r=0}^k \frac{(-1)^{r+1}B_{r+1}}{(r+1)!}(f^{(r)}(b) - f^{(r)}(a)) + \frac{(-1)^k}{(k+1)!} \int_a^b B_{k+1}(t)f^{(k+1)}(t)dt.$$
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"Euler-Maclaurin summation formula" is owned by KimJ.
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Cross-references: class, function, integer, Bernoulli periodic function, Bernoulli number
There are 3 references to this entry.
This is version 4 of Euler-Maclaurin summation formula, born on 2001-10-15, modified 2002-05-25.
Object id is 220, canonical name is EulerMaclaurinSummationFormula.
Accessed 11159 times total.
Classification:
| AMS MSC: | 65B15 (Numerical analysis :: Acceleration of convergence :: Euler-Maclaurin formula) |
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Pending Errata and Addenda
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