The Euler–Maclaurin formula is a powerful mathematical tool that provides a connection between discrete sums and continuous integrals. It is useful in various areas of numerical analysis, calculus, and asymptotic analysis. The formula allows us to approximate sums by integrals, compensating for the differences with correction terms.
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The Euler-Maclaurin formula relates a sum to an integral, endpoint corrections, and terms involving Bernoulli numbers and endpoint derivatives. Truncating after finitely many derivative corrections gives a remainder term that must be controlled; this supports accurate estimates of long sums.