Euler-Maclaurin Integration Formulas

The first Euler-Maclaurin integration formula is

$\int^1_0 f(x)\,dx = {\textstyle{1\over 2}}[f(1)+f(0)]$
$- \sum_{p=1}^q {1\over (2p)!} B_{2p}[f^{(2p-1)}(1)-f^{(2p-1)}(0)]$
$\mathop{+} {1\over (2q)!} \int^1_0 f^{(2q)}(x)B_{2q}\,dx,\quad$	(1)

where $B_n$ are Bernoulli Numbers. Sums may be converted to Integrals by inverting the Formula to obtain

$$\sum_{m=1}^n f(m) = \int^n_1 f(x)\,dx+{\textstyle{1\over 2}}[f(1)+f(n)]+{B_2\over 2!}[f'(n)-f'(1)]+\ldots.$$ (2)

For a more general case when $f(x)$ is tabulated at $n$ values $f_1$, $f_2$, ..., $f_n$,

$\int_{x_1}^{x_n} f(x)\,dx =h[{\textstyle{1\over 2}}f_1+f_2+f_3+\ldots+f_{n-1}+{\textstyle{1\over 2}}f_n]$
$\mathop{-}\sum_{k=1}^\infty {B_{2k}h^{2k}\over (2k)!} [{f_n}^{(2k-1)}-{f_1}^{(2k-1)}].\quad$	(3)

The Euler-Maclaurin formula is implemented in Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) as the function NSum with option Method->Integrate.

The second Euler-Maclaurin integration formula is used when $f(x)$ is tabulated at $n$ values $f_{3/2}$, $f_{5/2}$, ..., $f_{n-1/2}$:

$\int_{x_1}^{x_n} f(x)\,dx = h[f_{3/2}+f_{5/2}+f_{7/2}+\ldots+f_{n-3/2}+f_{n-1/2}]$
$-\sum_{k=1}^\infty {B_{2k}h^{2k}\over (2k)!}(1-2^{-2k+1})[{f_n}^{(2k-1)}-{f_1}^{(2k-1)}].$

See also Sum, Wynn's Epsilon Method

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 16 and 806, 1972.

Arfken, G. ``Bernoulli Numbers, Euler-Maclaurin Formula.'' §5.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 327-338, 1985.

Borwein, J. M.; Borwein, P. B.; and Dilcher, K. ``Pi, Euler Numbers, and Asymptotic Expansions.'' Amer. Math. Monthly 96, 681-687, 1989.

Vardi, I. ``The Euler-Maclaurin Formula.'' §8.3 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 159-163, 1991.