Maclaurin series examples
MaclaurinSeriesExamples
2025-07-14 22:03:52
2025-07-14 22:03:52
Example
Maclaurin series
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\subsection{Example 1}
One of the simplest Machlaurin series examples is the function
\begin{equation}
f\left(x\right) =\frac{1}{1-x} = \left(1-x\right)^{-1}
\end{equation}
Apply the chain rule to get the derivatives
$$f^{\prime}(x) = -\left(1-x\right)^{-2}(-1) = \frac{1}{\left(1-x\right)^2}$$
$$f^{\prime\prime}(x) = -2\left(1-x\right)^{-3}(-1) = \frac{2}{\left(1-x\right)^3}$$
$$f^{\prime\prime\prime}(x) = -6\left(1-x\right)^{-4}(-1) = \frac{6}{\left(1-x\right)^4}$$
Evaluating the function and its derivatives at zero yields
$$f(0) = \frac{1}{1-0} = 1$$
$$f^{\prime}(0) = \frac{1}{\left(1-0\right)^2} = 1$$
$$f^{\prime\prime}(0) = \frac{2}{\left(1-0\right)^3} = 2$$
$$f^{\prime\prime\prime}(0) = \frac{6}{\left(1-0\right)^4} = 6$$
Using the formula for the Machlaurin series
\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+\frac{f^{\prime\prime\prime}\left(0\right)}{3!}x^3+\dots
\end{equation}
\begin{equation}
f\left(x\right) = \sum _{m=0}^{\infty }\frac{f^{\left(m\right)}\left(0\right)}{m!}x^m=1+\frac{1}{1!}x+\frac{2}{2!}x^2+ \frac{6}{3!}x^3\dots
\end{equation}
Finally, evaluating the factorials and seeing they divide out the numerator we get the series
\begin{equation}
f\left(x\right) = 1+x+x^2+x^3\dots
\end{equation}
Let's evaluate example 1 function at $x=0.5$ and see how the order of the series (powers of x, so order 3 would go through $x^3$
\begin{figure}[h]
\centering
\resizebox{\textwidth}{!}{\includegraphics{example1.png}}
\caption{Example 1 at $x=0.5$}
\end{figure}
\begin{thebibliography}{9}
[1] Kreyszig, E., ``Advanced Engineering Mathematics.'' Fifth Edition. John Wiley and Sons, Inc. 1983. \\
\end{thebibliography}