## Euler-Maclaurin Integration Formulas

The first Euler-Maclaurin integration formula is
 (1)
where are Bernoulli Numbers. Sums may be converted to Integrals by inverting the Formula to obtain

 (2)

For a more general case when is tabulated at values , , ..., ,
 (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 , , ..., :

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.