# Euler-Maclaurin Summation Formula

## Contents

## Theorem

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

Then:

\(\displaystyle \sum_{k \mathop = 1}^{n - 1} f \left({k}\right)\) | \(=\) | \(\displaystyle \int_0^n f \left({x}\right) \rd x - \frac {f \left({0}\right) + f \left({n}\right)} 2 + \sum_{k \mathop = 1}^\infty \frac {B_{2 k} } {\left({2 k}\right)!} \left({f^{\left({2 k - 1}\right)} \left({n}\right) - f^{\left({2 k - 1}\right)} \left({0}\right)}\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_0^n f \left({x}\right) \rd x - \frac 1 2 \left({f \left({n}\right) + f \left({0}\right)}\right)\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {12} \left({F' \left({n}\right) - F' \left({0}\right)}\right)\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle - \, \) | \(\displaystyle \frac 1 {720} \left({F''' \left({n}\right) - F''' \left({0}\right)}\right)\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {30 \, 240} \left({F^{\left({5}\right)} \left({n}\right) - F^{\left({5}\right)} \left({0}\right)}\right)\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac 1 {1 \, 209 \, 600} \left({F^{\left({7}\right)} \left({n}\right) - F^{\left({7}\right)} \left({0}\right)}\right)\) |

where:

- $f^{\left({k}\right)}$ 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: $19.45$