Euler-Maclaurin Summation Formula

Theorem

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

Then:

$\ds \sum_{k \mathop = 1}^{n - 1} \map f k$

$=$

$\ds \int_0^n \map f x \rd x + \sum_{k \mathop = 1}^\infty \frac {B_k} {k!} \paren {\map {f^{\paren {k - 1} } } n - \paren {-1}^k \map {f^{\paren {k - 1} } } 0}$

$\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.

Proof

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Also known as

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

Also see

Cauchy Integral Test

Source of Name

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

Sources

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.6$ Infinite Series: Euler-Maclaurin Summation Formula: $3.6.28$
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
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Euler (or 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
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 21$: Series of Constants: The Euler-Maclaurin Summation Formula: $21.45.$