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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.3.67e
Chapter 11, Problem 11.3.67e

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The Taylor series for an even function centered at 0 has only even powers of x.

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Recall the definition of an even function: a function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain.
Consider the Taylor series of \( f(x) \) centered at 0, which is given by \( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \).
Substitute \( -x \) into the Taylor series: \( f(-x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-x)^n = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-1)^n x^n \).
Since \( f \) is even, \( f(-x) = f(x) \), so the series must satisfy \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-1)^n x^n = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \). This implies that terms with odd powers \( n \) must have zero coefficients because \( (-1)^n = -1 \) for odd \( n \), which would otherwise change the sign.
Therefore, the Taylor series for an even function centered at 0 contains only even powers of \( x \), confirming the statement is true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Even Functions

An even function satisfies f(-x) = f(x) for all x in its domain. This symmetry about the y-axis means the function's graph is mirrored on both sides, influencing the behavior of its Taylor series expansion.
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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. Centering at zero (Maclaurin series) expresses the function as powers of x, revealing patterns in coefficients based on the function's properties.
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Taylor Series

Parity of Terms in Taylor Series for Even Functions

For even functions centered at zero, all odd-powered terms in the Taylor series vanish because the derivatives of odd order at zero are zero. This results in a series containing only even powers of x, reflecting the function's symmetry.
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Taylor Series
Related Practice
Textbook Question

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

d. Suppose f'' is continuous on an interval that contains a, where f has an inflection point at a. Then the second−order Taylor polynomial for f at a is linear.

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Textbook Question

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

e. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?

Textbook Question

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

d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.

Textbook Question

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.


f. e⁻²ˣ


A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

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