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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 11.2.67c
Chapter 11, Problem 11.2.67c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).

Verified step by step guidance
1
Recall that the radius of convergence of a Taylor series centered at 0 is the distance from 0 to the nearest singularity of the function in the complex plane.
Given that the Taylor series of \( f \) converges only on the interval \( (-2, 2) \), this means the radius of convergence \( R \) of \( f \) is 2.
Now consider the function \( g(x) = f(x^2) \). To find the radius of convergence of the Taylor series of \( g \), analyze how the substitution \( x^2 \) affects the domain.
Since \( g(x) = f(x^2) \), the series for \( g \) converges when \( |x^2| < 2 \), which simplifies to \( |x| < \sqrt{2} \).
Therefore, the radius of convergence of the Taylor series for \( g(x) \) is \( \sqrt{2} \), which is different from the original radius 2, so the statement is false.

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Key Concepts

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

Radius of Convergence of a Taylor Series

The radius of convergence is the distance from the center point within which a Taylor series converges to the function. It depends on the function's behavior and singularities in the complex plane. Understanding this helps determine where the series representation is valid.
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Composition of Functions and Its Effect on Convergence

When composing functions, such as considering f(x²), the domain and convergence properties can change. The substitution x² affects the input range and may alter the radius of convergence of the resulting Taylor series.
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Counterexamples in Analysis

Counterexamples demonstrate that a general statement is false by providing a specific case where it fails. Constructing or identifying counterexamples is crucial to test claims about convergence and function behavior rigorously.
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Related Practice
Textbook Question

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


c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.

Textbook Question

Taylor series and interval of convergence


c. Determine the interval of convergence of the series.


f(x) = cosh 3x, a = 0

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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

d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?

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

c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

Textbook Question

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


d. 1/(1 + 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

Textbook Question

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


If f(x)=∑ₖ₌₀∞ cₖ xᵏ=0, for all x on an interval (−a, a), then cₖ = 0, for all k.

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