Editing Euler–Maclaurin formula (section)

===Approximation of integrals===
The formula provides a means of approximating a finite integral.  Let {{math|''a'' < ''b''}} be the endpoints of the interval of integration.  Fix {{mvar|N}}, the number of points to use in the approximation, and denote the corresponding step size by {{math|''h'' {{=}} {{sfrac|''b'' − ''a''|''N'' − 1}}}}.  Set {{math|''x<sub>i</sub>'' {{=}} ''a'' + (''i'' − 1)''h''}}, so that {{math|''x''<sub>1</sub> {{=}} ''a''}} and {{math|''x<sub>N</sub>'' {{=}} ''b''}}.  Then:<ref name="Devries">{{cite book |last1=Devries |first1=Paul L. |last2=Hasbrun |first2=Javier E. |title=A first course in computational physics. |edition=2nd |publisher=Jones and Bartlett Publishers |year=2011 |page=156}}</ref>
<math display=block>
\begin{align}
I & = \int_a^b f(x)\,dx \\
&\sim h\left(\frac{f(x_1)}{2} + f(x_2) + \cdots + f(x_{N-1}) + \frac{f(x_N)}{2}\right) + \frac{h^2}{12}\bigl[f'(x_1) - f'(x_N)\bigr] - \frac{h^4}{720}\bigl[f'''(x_1) - f'''(x_N)\bigr] + \cdots
\end{align}
</math>

This may be viewed as an extension of the [[trapezoid rule]] by the inclusion of correction terms.  Note that this asymptotic expansion is usually not convergent; there is some {{mvar|p}}, depending upon {{mvar|f}} and {{mvar|h}}, such that the terms past order {{mvar|p}} increase rapidly.  Thus, the remainder term generally demands close attention.<ref name="Devries"/>

The Euler–Maclaurin formula is also used for detailed [[error analysis (mathematics)|error analysis]] in [[numerical quadrature]]. It explains the superior performance of the [[trapezoidal rule]] on smooth [[periodic function]]s and is used in certain [[Series acceleration|extrapolation methods]]. [[Clenshaw–Curtis quadrature]] is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler–Maclaurin approach is very accurate (in that particular case the Euler–Maclaurin formula takes the form of a [[discrete cosine transform]]). This technique is known as a periodizing transformation.