Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=3ˣ, a=0
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.2.63a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
The interval of convergence of the power series ∑ cₖ(x−3)ᵏ could be (−2,8).
Verified step by step guidance1
Recall that the interval of convergence of a power series \( \sum c_k (x - a)^k \) is always centered at \( a \). In this problem, the series is centered at \( x = 3 \).
The interval of convergence is an interval \( (a - R, a + R) \), where \( R \) is the radius of convergence. Since the center is \( 3 \), the interval must be symmetric around 3.
Check if the given interval \( (-2, 8) \) is symmetric about 3. Calculate the distance from 3 to each endpoint: \( 3 - (-2) = 5 \) and \( 8 - 3 = 5 \). Both distances are equal, so the interval is symmetric about 3.
Since the interval \( (-2, 8) \) is symmetric about 3, it could represent the interval of convergence of the power series \( \sum c_k (x - 3)^k \) with radius of convergence \( R = 5 \).
Therefore, the statement is true because the interval \( (-2, 8) \) is a valid interval of convergence centered at 3 with radius 5.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series and Interval of Convergence
A power series is an infinite sum of the form ∑ cₖ(x−a)ᵏ centered at a point a. The interval of convergence is the set of x-values for which the series converges. This interval is always centered at a and extends symmetrically or asymmetrically around a, possibly including or excluding endpoints.
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08:44
Interval of Convergence
Radius of Convergence
The radius of convergence (R) is the distance from the center a to the endpoints of the interval of convergence. It determines how far from a the power series converges. The interval of convergence is (a−R, a+R), possibly including endpoints, so the length of the interval is always 2R.
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07:36Radius of Convergence
Testing Interval Endpoints and Counterexamples
To verify if a given interval can be the interval of convergence, check if it is centered at a and if its length matches twice the radius of convergence. If the interval is not symmetric about the center or does not match the radius, it cannot be the interval of convergence. Counterexamples help illustrate when intervals fail these conditions.
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05:44Divergence Test (nth Term Test)
Related Practice
Textbook Question
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
b. Find the radius and interval of convergence of the power series for J₀.
Textbook Question
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
a. Expand the integrand in a Taylor series centered at 0.
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = e²ˣ, a = 0
Textbook Question
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
a. Write out the first four terms of J₀.
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = (1 + x²)⁻¹, a = 0