Textbook Question
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
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 10.7.64e
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
Verified step by step guidance1
Recall that the radius of convergence \( R \) of a power series \( \sum a_n x^n \) is the distance from the center (here assumed to be 0) within which the series converges absolutely.
Given that the series converges at \( x = 4 \), this means \( |4| = 4 \) is within or on the boundary of the radius of convergence, so \( R \geq 4 \).
Given that the series diverges at \( x = 7 \), this means \( |7| = 7 \) is outside the radius of convergence, so \( R < 7 \).
Since \( R \) is between 4 and 7, the behavior of the series at \( x = 8 \) (where \( |8| = 8 \)) depends on whether 8 is inside or outside the radius of convergence.
Because \( 8 > R \), the series must diverge at \( x = 8 \). Therefore, the statement 'Diverges for \( x = 8 \)' is true.
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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 of a power series ∑ aₙxⁿ is the distance from the center (usually zero) within which the series converges absolutely. If the series converges at x = 4 but diverges at x = 7, the radius of convergence R satisfies 4 ≤ R < 7.
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Radius of Convergence
Behavior of Power Series Outside the Radius of Convergence
A power series diverges for all values of x whose absolute value exceeds the radius of convergence. Therefore, if |x| > R, the series must diverge. This helps determine convergence or divergence at points beyond the radius.
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07:36Radius of Convergence
Insufficient Information on Boundary Points
Convergence at points where |x| equals the radius of convergence is not guaranteed and must be checked separately. Since divergence at x = 7 implies R < 7, the behavior at x = 8 depends on whether 8 lies inside or outside the radius, which can be inferred but not always conclusively.
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04:50Critical Points
Related Practice
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 ]
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
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]