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Euler-Maclaurin summation formula (Theorem)

Let $ B_r$ be the $ r$th Bernoulli number, and $ B_r(x)$ be the $ r$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

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



"Euler-Maclaurin summation formula" is owned by KimJ.
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See Also: Bernoulli number

Keywords:  number theory

Attachments:
proof of Euler-Maclaurin summation formula (Proof) by pbruin
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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 9264 times total.

Classification:
AMS MSC: 65B15 (Numerical analysis :: Acceleration of convergence :: Euler-Maclaurin formula)
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