View source for MacLaurin formula

A particular case of the [[Taylor formula|Taylor formula]]. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620901.png" /> have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620902.png" /> derivatives at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620903.png" />. Then in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620904.png" /> of this point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620905.png" /> can be represented 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/m/m062/m062090/m0620906.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620907.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620908.png" />-th order remainder term, can be represented in some form or other.

The term  "MacLaurin formula"  is also used for functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620909.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209010.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209011.png" /> in the MacLaurin formula is taken to be a multi-index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209012.png" /> (see [[MacLaurin series|MacLaurin series]]). The formula is named after C. MacLaurin.



====Comments====
For some expressions for the remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209013.png" /> and for estimates of it see [[Taylor formula|Taylor formula]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 107–108</TD></TR></table>