Subject: Calculus

Euler Maclaurin Formula

Euler-Maclaurin Formula is a very important tool for giving an approximate for integrals in terms of finite sum or approximate finite sums in terms of integrals. Simply put, Euler-Maclaurin formula provides an accurate connection between integrals and sums. The general form of this formula can be written as

\sum_{n=a}^{b}f(n)=\int_{a}^{b}f(t)dt+\frac{1}{2}\left ( f(b)+f(a)\right )+\sum_{i=2}^{k}\frac{b_i}{i!}\left ( f^{i-1}(b)-f^{i-1}(a)\right )-\int_{a}^{b}\frac{B_{k}(1+t)}{k!}f^{k}(t)dt

where a and b are arbitrary real numbers with difference b – a > 0, Bn and bn are Bernoulli polynomials and numbers, and k is any positive integer. The real function f should have a continuous k-th derivative.

This formula was independently discovered by Leonhard Euler and Colin Maclaurin in the eighteenth century. However, neither of them was able to obtain the last term, R_k, of the formula which is very essential for the relation between sums and integrals.


Maclaurin’s approach was based on geometric structure while Euler used analytic ideas. Rk was later introduced by S. D. Poisson.
For a given function f(x), if f(x) and its derivatives tend to 0 as x approaches infinity, the Euler-Maclaurin formula can be simplified as follows by letting b \to \infty.

\sum_{n=a}^{\infty}f(n)=int_{a}^{\infty}f(t)dt+\frac{1}{2}\left (f(a)\right )+\sum_{i=2}^{k}\frac{b_i}{i!}\left ( f^{i-1}(b)-f^{i-1}(a)\right )-\int_{a}^{b}\frac{B_{k}(1+t)}{k!}f^{k}(t)dt

Consequently, for asymptotic series, the Euler-Maclaurin formula is written as

\sum_{n=a}^{b}f(n)\sim \int_{a}^{b}f(t)dt+\frac{1}{2}\left (f(b)+f(a)\right )+\sum_{k=1}^{k}\frac{b_{2k}}{2k!}\left ( f^{i-1}(b)-f^{2k-1}(a)\right )

where the symbol \sim indicates that the right-hand side is a so-called asymptotic series for the left hand side. This means that if we take the first n terms in the sum on the right-hand side, the error in approximating the left-hand side by that sum is at most on the order of the (n + 1)th term.

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