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 10.7.28c
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 ]
Verified step by step guidance1
Identify the given power series: \(\sum_{n=0}^{\infty} (-2)^n (n+1) (x-1)^n\).
Rewrite the series in a form that isolates the variable term: \(\sum_{n=0}^{\infty} (n+1) \left[-2(x-1)\right]^n\).
Use the Root Test or Ratio Test to find the radius of convergence. For the Ratio Test, consider the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\), where \(a_n = (n+1) \left[-2(x-1)\right]^n\).
Solve the inequality \(L < 1\) to find the interval of convergence in terms of \(x\).
Check the endpoints of the interval separately by substituting them back into the original series to determine if the series converges absolutely, conditionally, or diverges at those points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radius and Interval of Convergence
The radius of convergence determines the distance from the center point within which a power series converges absolutely. The interval of convergence includes all x-values for which the series converges, possibly including endpoints where convergence must be tested separately.
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Radius of Convergence
Absolute vs. Conditional Convergence
A series converges absolutely if the series of absolute values converges; otherwise, it may converge conditionally if the original series converges but not absolutely. Conditional convergence often occurs at the endpoints of the interval of convergence.
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07:51Choosing a Convergence Test
Ratio Test for Convergence
The ratio test uses the limit of the absolute value of the ratio of consecutive terms to determine convergence. If this limit is less than one, the series converges absolutely; if greater than one, it diverges; if equal to one, the test is inconclusive and further analysis is needed.
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05:35Ratio 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
A sequence of rational numbers is described as follows:
1/1,3/2,7/5,17/12,…,a/b,(a + 2b)/(a + b),…
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xₙ and yₙ be, respectively, the numerator and the denominator of the nᵗʰ fraction rₙ = xₙ / yₙ.
b. The fractions rₙ = xₙ / yₙ approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that rₙ² − 2 = ±(1 / yₙ)² and that yₙ is not less than n.)
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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
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 ]