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 10.7.32c
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) ]
Verified step by step guidance1
Rewrite the series to identify its general term clearly: \( a_n = \frac{(3x + 1)^{n+1}}{2n + 2} \). This helps in analyzing the convergence behavior.
Determine the radius and interval of convergence by applying the Ratio Test. Compute \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) and simplify the expression to find the values of \( x \) for which the limit is less than 1.
Find the interval of convergence from the inequality obtained in the Ratio Test. This gives the open interval where the series converges absolutely.
Check the endpoints of the interval separately by substituting them back into the original series. Since the question asks for conditional convergence, test whether the series converges at these endpoints but not absolutely.
Use appropriate convergence tests (such as the Alternating Series Test or the p-series test) on the series at the endpoints to determine if the convergence is conditional there.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Interval of Convergence
The interval of convergence is the set of all x-values for which a given power series converges. To find it, one typically uses the Ratio or Root Test to determine the radius of convergence, then checks the endpoints separately to see if the series converges there.
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08:44
Interval of Convergence
Conditional vs. Absolute Convergence
A series converges absolutely if the series of absolute values converges; otherwise, if the series converges but not absolutely, it converges conditionally. Conditional convergence often occurs at the endpoints of the interval of convergence and requires tests like the Alternating Series Test.
Recommended video:
07:51Choosing a Convergence Test
Convergence Tests for Series
Tests such as the Ratio Test, Root Test, and Alternating Series Test help determine whether a series converges or diverges. The Ratio Test is useful for power series, while the Alternating Series Test can confirm conditional convergence when terms alternate in sign and decrease in magnitude.
Recommended video:
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 = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
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 ]
Textbook Question
Intervals of Convergence
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]