Let a<b be integers and f(x) be a function on [a,b] with derivatives of all orders. Let $I={\int }_a^{b}f\left(x\right)\text{d}x$ and let S denote the estimate applying the trapezium rule to this integral using unit intervals. The Euler-Maclaurin formula then states

where B_n denotes the nth Bernoulli number, and provided that

See also Bernoulli polynomials.