View source for Euler-MacLaurin formula

A summation formula that connects the partial sums of a series with the integral and derivatives of its general term:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365201.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365202.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365203.png" /> are the [[Bernoulli numbers|Bernoulli numbers]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365204.png" /> is the remainder. Using the [[Bernoulli polynomials|Bernoulli polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365206.png" />, the remainder can be rewritten in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365207.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365208.png" /> the remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e0365209.png" /> can be expressed by means of the Bernoulli numbers:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652010.png" /></td> </tr></table>

If the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652012.png" /> have the same sign and do not change sign on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652013.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652014.png" /></td> </tr></table>

If, furthermore,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652015.png" /></td> </tr></table>

then the Euler–MacLaurin formula becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652016.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652017.png" /></td> </tr></table>

This version is used, for example, to derive the [[Stirling formula|Stirling formula]], in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652019.png" /> is the [[Euler constant|Euler constant]]. The formula has also been generalized to multiple sums.

The Euler–MacLaurin formula finds application in the approximate calculation of definite integrals, the study of convergence of series, the computation of sums, and the expansion of functions in Taylor series. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652023.png" />, it yields the expression

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652024.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652025.png" /></td> </tr></table>

The Euler–MacLaurin formula plays an important role in the study of asymptotic expansions, number-theoretic estimates and [[Finite-difference calculus|finite-difference calculus]].

Sometimes the Euler–MacLaurin formula is applied in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652026.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652027.png" /></td> </tr></table>

The formula was first obtained by L. Euler [[#References|[1]]] as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652029.png" /> is the sum of the first terms of the series with general term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652032.png" />, and the coefficients are determined from the recurrence relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652033.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036520/e03652034.png" /></td> </tr></table>

The formula was later discovered independently by C. MacLaurin [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,   ''Comment. Acad. Sci. Imp. Petrop.'' , '''6'''  (1738)  pp. 68–97</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. MacLaurin,   "A treatise of fluxions" , '''1–2''' , Edinburgh  (1742)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Nörlund,   "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.O. [A.O. Gel'fond] Gelfond,   "Differenzenrechnung" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR></table>



====Comments====
The use of the Euler–MacLaurin sum formula in numerical [[Quadrature|quadrature]] is discussed in [[#References|[a1]]] and [[#References|[a2]]]. By replacing the various derivatives by finite differences the quadrature rules of Bessel, Gauss and Gregory are obtained.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,   "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.F. Steffensen,   "Interpolation" , Chelsea, reprint  (1950)</TD></TR></table>