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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.1.65a
Capitolo 11, Problema 11.1.65a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. Only even powers of x appear in the Taylor polynomials for f(x)=e⁻²ˣ centered at 0.

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Recall that the Taylor polynomial of a function \(f(x)\) centered at 0 (Maclaurin series) is given by the sum \(\displaystyle \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n\), where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
Consider the function \(f(x) = e^{-2x}\). Its Maclaurin series expansion can be found by substituting \(-2x\) into the exponential series \(e^t = \sum_{n=0}^\infty \frac{t^n}{n!}\), giving \(e^{-2x} = \sum_{n=0}^\infty \frac{(-2x)^n}{n!} = \sum_{n=0}^\infty \frac{(-2)^n}{n!} x^n\).
Notice that the series contains terms for all powers of \(x\), both even and odd, because \(n\) runs over all nonnegative integers and the coefficient \(\frac{(-2)^n}{n!}\) is nonzero for every \(n\).
Since the Taylor polynomial includes terms with odd powers of \(x\) (like \(x^1\), \(x^3\), etc.) with nonzero coefficients, it is not true that only even powers of \(x\) appear in the Taylor polynomials for \(f(x) = e^{-2x}\) centered at 0.
Therefore, the statement is false, and a counterexample is the first-degree term in the expansion, which is \(\frac{(-2)^1}{1!} x = -2x\), an odd power term.

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Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. For a function f(x) centered at 0, the series is f(x) = Σ (f⁽ⁿ⁾(0)/n!) xⁿ, where n is a non-negative integer. Understanding how to find these derivatives and form the series is essential to analyze the polynomial terms.
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Taylor Series

Even and Odd Powers in Series

The presence of only even or odd powers in a Taylor series depends on the function's symmetry. Even functions satisfy f(-x) = f(x) and have Taylor expansions with only even powers, while odd functions satisfy f(-x) = -f(x) and have only odd powers. Recognizing the function's parity helps predict the powers appearing in its series.
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Intro to Power Series

Exponential Function and Its Derivatives

The exponential function e^u, where u is a function of x, has derivatives that follow a predictable pattern. For f(x) = e^{-2x}, each derivative involves powers of -2 and e^{-2x}. Evaluating these at 0 helps determine the coefficients in the Taylor series and whether odd or even powers appear.
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Derivatives of General Exponential Functions
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