Use a Maclaurin series in Table 1 to obtain the Maclaurin...

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.

$ f(x) = \left\{\begin{array} (\frac{x -\sin x}{x^3} \text{ if } x \not= 0 \\\\ \frac{1}{6} \text { if } x = 0 \end{array}\right. $

Key Concepts

Maclaurin Series

The Maclaurin series is a Taylor series expansion of a function about x=0. It expresses a function as an infinite sum of powers of x, where each coefficient is derived from the function's derivatives at 0. This series is a crucial tool for approximating functions and studying their behavior near the origin.

Series Expansion of Trigonometric Functions

Trigonometric functions such as sine and cosine have well-known Maclaurin series expansions. For example, the sine function can be expanded as an infinite sum involving alternating powers of x divided by factorials. These expansions are essential for deriving series representations for functions that involve trigonometric components.

Algebraic Manipulation of Series

Once individual series are known, algebraic manipulation—such as subtracting one series from another, dividing, or multiplying series term-by-term—allows the derivation of new series representations for composite functions. This process is key when combining known Maclaurin series to obtain the series for more complex expressions.

Continuity in Piecewise-Defined Functions

For functions defined piecewise, ensuring continuity often involves matching a value at a specific point with the limit of the function’s series expansion as x approaches that point. This technique guarantees that the function is well-behaved and continuous, making the series representation valid across its entire domain.

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