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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 11.2.63d
Chapter 11, Problem 11.2.63d

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.

Verified step by step guidance
1
Recognize that the function \( f(x) = \sum_{k=0}^{\infty} c_k x^k \) is represented as a power series centered at 0, which converges on the interval \((-a, a)\).
Recall the uniqueness theorem for power series: if a power series converges to zero for all \( x \) in an interval around zero, then all its coefficients must be zero.
Understand that this theorem follows from the fact that power series are analytic functions, and an analytic function that is identically zero on an interval must have all zero coefficients in its power series expansion.
To see why, consider differentiating \( f(x) \) term-by-term and evaluating at \( x=0 \). The \( k \)-th derivative at zero is related to \( c_k \) by \( f^{(k)}(0) = k! c_k \). Since \( f(x) = 0 \) for all \( x \), all derivatives at zero must be zero, implying \( c_k = 0 \) for all \( k \).
Therefore, the statement is true: if \( f(x) = 0 \) for all \( x \) in \((-a, a)\), then all coefficients \( c_k \) must be zero.

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Key Concepts

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

Power Series and Radius of Convergence

A power series is an infinite sum of the form ∑ cₖ xᵏ, which converges within a certain radius around zero called the radius of convergence. Within this interval, the series represents a function that can be analyzed term-by-term. Understanding convergence is essential to determine where the series equals zero.
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Radius of Convergence

Uniqueness Theorem for Power Series

If a power series converges to zero for all x in an interval around zero, then all its coefficients must be zero. This theorem ensures that the zero function has a unique power series representation, meaning no nontrivial power series can vanish identically on an interval without all coefficients being zero.
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Intro to Power Series

Counterexamples and Zero Functions

To verify the statement, one must consider whether a nonzero power series can equal zero on an interval. Counterexamples involve functions that vanish at points but not on an entire interval. The concept clarifies that if the function is zero on an interval, the coefficients must be zero, reinforcing the uniqueness of power series expansions.
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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

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).

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

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

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.


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