What is the Maclaurin Summation Formula?
The Euler–Maclaurin summation formula, discovered independently by Leonhard Euler (in 1732) and Colin Maclaurin (in 1742), relates the summation of a function to an integral approximation. It gives a way to calculate “corrections” in terms of the function’s derivatives, evaluated at the endpoints. It’s most general form is :
- a & b are arbitrary real numbers; a – b is a positive integer,
- Bn is a Bernoulli polynomial,
- bn is a Bernoulli number,
- k = a positive integer.
In addition, the function f must have a continuous k
If the function and all of its derivatives approach zero as x→∞, the Euler–Maclaurin summation Formula simplifies (by letting b→∞ in the identity) to:
It can also be expressed in a more “modern” asymmetrical form as :
With a remainder, at any stage, of
The Euler-Maclaurin summation formula has many uses in mathematics and science, including quantum topology, where the formula is sometimes needed for a resurgent function with singularities in the vertical strip perpendicular to the range of summation . It is also one way to generate an asymptotic series.
What are Bernoulli Numbers?
Bernoulli numbers are rational numbers defined as coefficients in the series expansion
These coefficients are very challenging to calculate by hand; software is normally used to find them .
Bernoulli numbers Bk are defined as values of the Bernoulli polynomials Bk(t) at t = 0: Bk = Bk(0). Bernoulli polynomials satisfy the differential equation 
B′k+1(t) = (k + 1)Bk(t), with B2k+1(0) = 0, B2k+1(1) = 0, for all k > 0.
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Vol. 33, No. 1/3 (1985), pp. 1-13 (13 pages). Springer.
 Costin, O. & Garoufaldis, S. Resurgence of the E-M Summation Formula. Retrieved August 14, 2021 from: https://people.math.osu.edu/costin.9/EULMCL.pdf
 Rozman, M. (2016). Euler-Maclaurin summation formula. Retrieved August 14, 2021 from: https://www.phys.uconn.edu/~rozman/Courses/P2400_16S/downloads/euler-maclaurin-summation.pdf
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