EulerMaclaurin

The Euler–Maclaurin formula is a fundamental result in analysis that relates a finite sum to an integral, providing a precise connection between discrete sums and continuous quantities. It is named after Leonhard Euler and Colin Maclaurin, who developed variant forms of the idea in the 18th century. The formula is widely used to approximate sums, derive asymptotic expansions, and improve numerical integration by incorporating correction terms.

Standard finite form: For a function f with sufficient smoothness on [a,b], the sum over integers satisfies

sum_{n=a}^b f(n) = ∫_a^b f(x) dx + (f(a) + f(b))/2 + sum_{k=1}^{m} B_{2k}/(2k)! [f^{(2k-1)}(b) - f^{(2k-1)}(a)] + R_m,

where B_{2k} are Bernoulli numbers and R_m is a remainder term that depends on higher derivatives of f. A common specialization is to take a = 1, which yields a widely used expansion for partial sums.

Variants and conditions: There are multiple formulations, including infinite-series forms and versions using periodic Bernoulli polynomials. The remainder term can be bounded under smoothness assumptions, and the choice of m governs the accuracy of the approximation.

Applications and significance: The Euler–Maclaurin formula is instrumental in numeric analysis for accelerating convergence of series, estimating harmonic numbers and zeta-function values, and refining the trapezoidal rule in numerical integration. It also provides insights in analytic number theory by linking discrete sums to integrals and derivatives.

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