MacLaurin series

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)