# Euler-Maclaurin Summation Formula

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## 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.

## Proof

## Also see

## Source of Name

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

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 19$: The Euler-Maclaurin Summation Formula: $19.45$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Euler-Maclaurin summation formula**