The Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.
If n is a natural number and f(x) is a smooth (meaning: sufficiently often differentiable) function defined for all real numbers x between 0 and n, then the integral
R is an error term which is normally small if p is large enough and can be estimated as
If f is a polynomial and p is big enough, then the remainder term vanishes. For instance, if f(x) = x3, we can choose p = 2 to obtain after simplification
External links: