The first Euler-Maclaurin integration formula is
(1) |
(2) |
(3) |
The Euler-Maclaurin formula is implemented in Mathematica (Wolfram Research, Champaign, IL) as the function NSum with option Method->Integrate.
The second Euler-Maclaurin integration formula is used when is tabulated at values , , ..., :
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.