Editing Euler–Maclaurin formula (section)

{{Short description|Summation formula}}
{{Use American English|date = March 2019}}
In [[mathematics]], the '''Euler–Maclaurin formula''' is a formula for the difference between an [[integral]] and a closely related [[Summation|sum]].  It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and [[series (mathematics)|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.  Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.  It was later generalized to [[Darboux's formula]].