power seriesPowerSeries2025-07-13 17:15:112025-07-13 20:47:25Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands hereIn mathematics, a power series (in one variable) is an infinite series of the form
\begin{equation}
\sum _{m=0}^{\infty }c_{m}\left(x-a\right)^{m}=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}+\dots
\end{equation}
where $c_{m}$ represents the coefficient of the $m$th term and $a$ is a constant called the center of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. \\
In many situations, the center $a$ is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form
\begin{equation}
\sum _{m=0}^{\infty }c_{m}\left(x)^{m}=c_{0}+c_{1}x+c_{2}x^{2}+\dots
\end{equation}