Euler–Maclaurin formula

The Euler–Maclaurin formula relates a finite sum of function values to an integral, endpoint values, and higher derivatives, with a remainder term.

Bernoulli numbers: A sequence of rational numbers that appears in power-sum formulas and many series expansions. They supply the coefficients of the formula’s endpoint-derivative corrections.
Stirling's approximation: An asymptotic formula approximating factorials using powers and an exponential function. Applying the formula to the logarithm of a factorial yields its correction series.
Leonhard Euler: A Swiss mathematician whose work shaped analysis, number theory, mechanics, and mathematical notation. Euler developed summation formulas that became one source of the result.
Poisson summation formula: An identity relating a sum of a function over integers to a sum involving its Fourier transform. It connects sums and integrals through frequency space rather than endpoint derivatives.
Asymptotic series: A formal series whose finite truncations approximate a quantity in a limiting regime, without requiring convergence. Euler–Maclaurin corrections can grow after an initially accurate sequence of truncations.
Bernoulli polynomials: A sequence of polynomials generated from Bernoulli numbers and used in summation and integration formulas. Their periodic versions express the formula’s remainder.
Harmonic number: The nth harmonic number is the sum of reciprocals from one through n. The formula produces its logarithmic growth and finite-size corrections.
Colin Maclaurin: A Scottish mathematician known for work on series expansions and the calculus of variations. Maclaurin independently developed a form of the summation formula.
Abel–Plana formula: A summation formula expressing sums of analytic functions through integrals and contour-related corrections. It is an alternative sum-to-integral relation, especially for analytic functions.
Optimal truncation: The practice of stopping an asymptotic expansion where its terms are smallest. It can extract the best approximation before later corrections increase the error.

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