Euler–Maclaurin formula

In mathematics, 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. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735 (and later generalized as Darboux's formula). Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.

If $m$ and $n$ are natural numbers and $f (x)$ is a complex or real valued continuous function for real numbers $x$ in the interval $[m, n]$ , then the integral

can be approximated by the sum (or vice versa)

(see trapezoidal rule). The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives $f (k) (x)$ evaluated at the end points of the interval, that is to say when $x = m$ and $x = n$ .

Explicitly, for a natural number $p$ and a function $f (x)$ that is $p$ times continuously differentiable in the interval $[m, n]$ , we have

where $B k$ is the $k$ th Bernoulli number (with $B 1 = 1 / 2$ ) and $R$ is an error term which is normally small for suitable values of $p$ and depends on $n$ , $m$ , $p$ , and $f$ .

...
Wikipedia