Find the Taylor series at x=0 of the given function. Use...

Find the Taylor series at $x=0$ of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at $x=0$ of $\frac{1}{1-x}, e^{x},$ or $\cos x .$ These series are derived in Examples 1 and 2 and Check Your Understanding Problem 2. $$\sin 3 x$$

Key Concepts

Series Expansion of Trigonometric Functions

The power series expansion of trigonometric functions, such as sine and cosine, is a fundamental example of Taylor series. These expansions are derived from calculating successive derivatives of the trigonometric function at a specific point, typically zero, and are essential in approximating periodic functions or functions involving oscillatory behavior.

Substitution in Power Series

Substitution in power series involves replacing the variable in a known series expansion with another expression, such as scaling the variable by a constant factor. This technique allows one to derive series for more complex functions by applying appropriate substitutions to simpler, well-known series.

Taylor Series

The Taylor series is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It provides a powerful tool for approximating functions near that point and for performing operations like differentiation and integration term-by-term on the power series.

Maclaurin Series

A Maclaurin series is a special type of Taylor series that is centered at x = 0. By expressing functions as Maclaurin series, one can often derive simpler forms of complex functions and perform substitutions or manipulations efficiently to obtain series expansions for related functions.

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