Category:Euler-Maclaurin Summation Formula
Jump to navigation
Jump to search
This category contains pages concerning Euler-Maclaurin Summation Formula:
Let $f$ be a real function which is appropriately differentiable and integrable.
Then:
| \(\ds \sum_{k \mathop = 1}^{n - 1} \map f k\) | \(=\) | \(\ds \int_0^n \map f x \rd x + \sum_{k \mathop = 1}^\infty \frac {B_k} {k!} \paren {\map {f^{\paren {k - 1} } } n - \paren {-1}^k \map {f^{\paren {k - 1} } } 0}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^n \map f x \rd x\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac 1 2 \paren {\map f n + \map f 0}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {12} \paren {\map {f'} n - \map {f'} 0}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac 1 {720} \paren {\map {f' ' '} n - \map {f' ' '} 0}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 {30 \, 240} \paren {\map {f^{\paren 5} } n - \map {f^{\paren 5} } 0}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac 1 {1 \, 209 \, 600} \paren {\map {f^{\paren 7} } n - \map {f^{\paren 7} } 0}\) |
where:
- $f^{\paren k}$ denotes the $k$th derivative of $f$
- $B_n$ denotes the $n$th Bernoulli number.
Source of Name
This entry was named for Leonhard Paul Euler and Colin Maclaurin.
Pages in category "Euler-Maclaurin Summation Formula"
The following 3 pages are in this category, out of 3 total.