Textbook Question
Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 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 11.2.15
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (x/3)ᵏ
Verified step by step guidance1
Identify the given power series: \( \sum_{k=0}^{\infty} \left( \frac{x}{3} \right)^k \). This is a geometric series with the common ratio \( r = \frac{x}{3} \).
Recall that a geometric series \( \sum_{k=0}^{\infty} r^k \) converges if and only if \( |r| < 1 \). Apply this condition to the given series: \( \left| \frac{x}{3} \right| < 1 \).
Solve the inequality \( \left| \frac{x}{3} \right| < 1 \) to find the interval of convergence. Multiply both sides by 3 to get \( |x| < 3 \). This means the radius of convergence \( R = 3 \).
Express the interval of convergence as \( (-3, 3) \). Next, check the endpoints \( x = -3 \) and \( x = 3 \) by substituting them back into the series to determine if the series converges at these points.
At \( x = 3 \), the series becomes \( \sum_{k=0}^{\infty} 1^k \), which diverges. At \( x = -3 \), the series becomes \( \sum_{k=0}^{\infty} (-1)^k \), which also diverges. Therefore, the interval of convergence is \( (-3, 3) \) without including the endpoints.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series
A power series is an infinite sum of terms in the form a_k(x - c)^k, where a_k are coefficients and c is the center. Understanding power series involves recognizing how the variable x affects convergence depending on its distance from the center.
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Intro to Power Series
Radius of Convergence
The radius of convergence is the distance from the center within which a power series converges absolutely. It can be found using tests like the Ratio Test, and it defines the interval where the series behaves well.
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07:36Radius of Convergence
Interval of Convergence
The interval of convergence is the set of x-values for which the power series converges. It includes all points within the radius of convergence and requires checking endpoints separately to determine if they are included.
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08:44Interval of Convergence
Related Practice
Textbook Question
{Use of Tech} Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with n = 3.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
e⁰ᐧ¹²
Textbook Question
Suppose a power series converges if |x−3|<4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.
Textbook Question
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
g(x) = − 1/(1 + x)² using f(x) = 1/(1 + x)
Textbook Question
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (sin 2x)/x
Textbook Question
{Use of Tech} Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with n = 3.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
√1.06