View source for MacLaurin series

''for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621001.png" />''

The power series

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It was studied by C. MacLaurin [[#References|[1]]]. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621003.png" /> analytic at zero is expanded as a [[Power series|power series]] around zero, then this series coincides with the MacLaurin series. When a function depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621004.png" /> variables, the MacLaurin series is a multiple power series:

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in which the summation is over the multi-indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621007.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621008.png" /> are non-negative integers. A MacLaurin series is a special case of a [[Taylor series|Taylor series]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. MacLaurin,   "A treatise of fluxions" , '''1–2''' , Edinburgh  (1742)</TD></TR></table>