MacLaurin series
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
for a function $f(z)$
The power series
$$f(z)=\sum_{k=0}^\infty\frac{f^{(k)}(0)}{k!}z^k.$$
It was studied by C. MacLaurin [1]. If a function $f(z)$ analytic at zero is expanded as a power series around zero, then this series coincides with the MacLaurin series. When a function depends on $m$ variables, the MacLaurin series is a multiple power series:
$$\sum_{|k|=0}^\infty\frac{f^{(k_1)}(0)\dots f^{(k_m)}(0)}{k_1!\dots k_m!}z_1^{k_1}\dots z_m^{k_m}$$
in which the summation is over the multi-indices $k=(k_1,\dots,k_m)$, $|k|=|k_1|+\dots+|k_m|$, and $k_i$ are non-negative integers. A MacLaurin series is a special case of a Taylor series.
References
[1] | C. MacLaurin, "A treatise of fluxions" , 1–2 , Edinburgh (1742) |
How to Cite This Entry:
MacLaurin series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MacLaurin_series&oldid=43457
MacLaurin series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MacLaurin_series&oldid=43457
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article