Textbook Question
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.7.65f
Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
f. Diverges for x = 4.9
Verified step by step guidance1
Identify the center of the power series, which is at \(x = 2\), since the series is given as \(\sum a_n (x - 2)^n\).
Use the information about convergence and divergence at specific points to find the radius of convergence \(R\). The series converges at \(x = -1\) and diverges at \(x = 6\).
Calculate the distance from the center to these points: \(|-1 - 2| = 3\) and \(|6 - 2| = 4\). Since the series converges at \(x = -1\) (distance 3) and diverges at \(x = 6\) (distance 4), the radius of convergence \(R\) satisfies \(3 \leq R < 4\).
Determine whether the series converges or diverges at \(x = 4.9\) by calculating the distance from the center: \(|4.9 - 2| = 2.9\).
Since \(2.9 < 3 \leq R\), the point \(x = 4.9\) lies within the radius of convergence, so the series converges at \(x = 4.9\). Therefore, the statement that it diverges at \(x = 4.9\) 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
The radius of convergence is the distance from the center of a power series within which the series converges absolutely. For a series centered at x = 2, it converges for all x such that |x - 2| < R, where R is the radius. Knowing convergence and divergence at specific points helps estimate R.
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Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all x-values for which the power series converges. It is centered at the series' center (here, x = 2) and extends R units in both directions. Convergence at boundary points must be checked separately, as behavior can differ.
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08:44Interval of Convergence
Testing Convergence at Specific Points
To determine if a series converges or diverges at a particular x, compare the distance |x - center| to the radius of convergence. If the distance is less than R, the series converges; if greater, it diverges. If equal, convergence must be tested individually.
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07:51Choosing a Convergence Test
Related Practice
Textbook Question
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
Textbook Question
Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
e. Diverges for x = 8