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.

R_k=\int_{a}^{b}\frac{B_{k}(1+t)}{k!}f^{k}(t)dt

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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