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.

This article needs proofreading. In particular: Confusion between the various types of Bernoulli numbers means I may have got signs and indices wrong. If you believe all issues are dealt with, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Proof

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Also known as

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

Also see

• Integral Test

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: $19.45$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Euler-Maclaurin summation formula
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler-Maclaurin summation formula
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler-Maclaurin summation formula