Maclaurin seriesMaclaurinSeries2025-07-13 20:34:002025-07-13 20:42:56Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands hereA Taylor series is also called a Maclaurin series when $0$ is the point wherethe derivatives are considered. It is named after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The Taylor series of a real or complex-valued function $f\left(x\right)$, that is infinitely differentiable at a real or complex number $a$, is the power series
\begin{equation}
f\left(x\right) = \sum _{m=0}^{\infty }\frac{f^{\left(m\right)}\left(a\right)}{m!}\left(x-a\right)^m=f\left(a\right)+\frac{f^{\prime}\left(a\right)}{1!}\left(x-a\right)+\frac{f^{\prime\prime}\left(a\right)}{2!}\left(x-a\right)^2+\dots
\end{equation}
Here, $m!$ denotes the factorial of $m$. The function $f^{\left(m\right)}\left(a\right)$ denotes the $m$th derivative of $f$ evaluated at the point $a$. The derivative of order zero of $f$ is defined to be $f$ itself and $\left(x-a\right)^0$ and $0!$ are both defined to be $1$. With $a=0$, the Maclaurin series takes the form: \\
\begin{equation}
f\left(x\right) = \sum _{m=0}^{\infty }\frac{f^{\left(m\right)}\left(0\right)}{m!}x^m=f\left(0\right)+\frac{f^{\prime}\left(0\right)}{1!}x+\frac{f^{\prime\prime}\left(0\right)}{2!}x^2+\dots
\end{equation}
This article is a derivative work of the creative commons share alike with attribution in [1].
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[1] Wikipedia contributors, ``Taylor series,'' Wikipedia, The Free Encyclopedia. Accessed July 13, 2025. \\
[2] Kreyszig, E., ``Advanced Engineering Mathematics.'' Fifth Edition. John Wiley and Sons, Inc. 1983. \\
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