MacLaurin series

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for a function $f(z)$

The power series


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:

$$\sum_{|k|=0}^\infty\frac{f^{(k_1)}(0)\cdots f^{(k_m)}(0)}{k_1!\cdots k_m!}z_1^{k_1}\cdots z_m^{k_m}$$

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.


[1] C. MacLaurin, "A treatise of fluxions" , 1–2 , Edinburgh (1742)
How to Cite This Entry:
MacLaurin series. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article