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. $$3\left(e^{-2 x}-2\right)$$

Key Concepts

Taylor Series Expansion

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point, usually around x=0. This series approximates the function and provides insight into its behavior near that point by expressing it in a polynomial form.

Power Series of the Exponential Function

The power series expansion of the exponential function, such as e^x, serves as a standard example in calculus. It is derived by evaluating the derivatives of e^x at the expansion point, leading to the series ? (x^n/n!) for n from 0 to infinity. This series converges for all real numbers and can be modified for variations of the exponential function.

Substitution in Series

Substitution in power series involves replacing the variable x in a known series with another expression, which allows the examination of functions like e^(ax) for any constant a. This is a crucial technique for adapting standard series formulas to similar functions that differ by a scaling or translation of the variable.

Series Manipulation

Series manipulation encompasses operations such as multiplication by constants, addition, subtraction, or re-indexing of series. This enables one to build new series representations from known series, such as subtracting a constant term or scaling the series by a factor, while preserving the convergence properties and revealing information about the function's behavior.

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