Euler–Maclaurin formula

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In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Euler–Maclaurin formula is a formula for the difference between an <a href="/facts/Integral/V7lyV997">integral</a> and a closely related <a href="/facts/Summation/V2WUxoKn">sum</a>.  It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and <a href="/facts/Series_(mathematics)/yDnlsgIe">infinite series</a> using integrals and the machinery of <a href="/facts/Calculus/MEmUTYoz">calculus</a>. For example, many asymptotic expansions are derived from the formula, and <a href="/facts/Faulhaber%2527s_formula/bQJhbEnp">Faulhaber's formula</a> for the sum of powers is an immediate consequence.
</p><p>The formula was discovered independently by <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> and <a href="/facts/Colin_Maclaurin/nVexkojA">Colin Maclaurin</a> around 1735.  Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.  It was later generalized to <a href="/facts/Darboux%2527s_formula/fgqKU3aE">Darboux's formula</a>.
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Euler–Maclaurin formula open-in-new

Euler–Maclaurin formula