Euler-Maclaurin Summation Formula

Theorem

Let $f$ be a real function which is appropriately differentiable and integrable.

Then:

 $\ds \sum_{k \mathop = 1}^{n - 1} \map f k$ $=$ $\ds \int_0^n \map f x \rd x - \frac {\map f 0 + \map f n} 2 + \sum_{k \mathop = 1}^\infty \frac {B_{2 k} } {\paren {2 k}!} \paren {\map {f^{\paren {2 k - 1} } } n - \map {f^{\paren {2 k - 1} } } 0}$ $\ds$ $=$ $\ds \int_0^n \map f x \rd x - \frac 1 2 \paren {\map f n + \map f 0}$ $\ds$  $\, \ds + \,$ $\ds \frac 1 {12} \paren {\map {f'} n - \map {f'} 0}$ $\ds$  $\, \ds - \,$ $\ds \frac 1 {720} \paren {\map {f'''} n - \map {f'''} 0}$ $\ds$  $\, \ds + \,$ $\ds \frac 1 {30 \, 240} \paren {\map {f^{\paren 5} } n - \map {f^{\paren 5} } 0}$ $\ds$  $\, \ds - \,$ $\ds \frac 1 {1 \, 209 \, 600} \paren {\map {f^{\paren 7} } n - \map {f^{\paren 7} } 0}$

where:

$f^{\paren k}$ denotes the $k$th derivative of $f$
$B_n$ denotes the $n$th Bernoulli number.

Also known as

The Euler-Maclaurin Summation Formula is also seen referred to as the Euler Summation Formula.

Source of Name

This entry was named for Leonhard Paul Euler and Colin Maclaurin.