## 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 [1]:

Where:

- a & b are arbitrary real numbers; a – b is a positive integer,
- B
_{n}is a Bernoulli polynomial, - b
_{n}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 [2]:

With a remainder, at any stage, of

and

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 [3]. 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 [4].

Bernoulli numbers B_{k} are defined as values of the Bernoulli polynomials B_{k}(t) at t = 0: B_{k} = B_{k}(0). Bernoulli polynomials satisfy the differential equation [5]

B′_{k+1}(t) = (k + 1)B_{k}(t), with B_{2k+1}(0) = 0, B_{2k+1}(1) = 0, for all k > 0.

## References

[1] Kac, V. (2005). 18.704 Seminar in Algebra and Number Theory Fall 2005. Euler-Maclaurin Formula. Retrieved August 14, 2021 from: http://people.csail.mit.edu/kuat/courses/euler-maclaurin.pdf

[2]Mills, S. (1985). The Independent Derivations by Leonhard Euler and Colin MacLaurin of the Euler-MacLaurin Summation Formula. Archive for History of Exact Sciences

Vol. 33, No. 1/3 (1985), pp. 1-13 (13 pages). Springer.

[3] 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

[4] 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

[5] Verschelde, J. The Euler-Maclaurin Summation Formula. Retrieved August 14, 2021 from:

http://homepages.math.uic.edu/~jan/MCS471/Lec28/emsum.pdf