Textbook Question
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=2 to ∞) 1 / ln n
(Hint: Limit Comparison with ∑ (from n=2 to ∞) (1/n))
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.7.6
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) (2x)ⁿ
Verified step by step guidance1
Identify the given series: \( \sum_{n=0}^{\infty} (2x)^n \). This is a geometric series with common ratio \( r = 2x \).
Recall that a geometric series \( \sum r^n \) converges if and only if \( |r| < 1 \). So, set up the inequality \( |2x| < 1 \) to find the interval of convergence.
Solve the inequality \( |2x| < 1 \) to get \( |x| < \frac{1}{2} \). This gives the radius of convergence \( R = \frac{1}{2} \) and the open interval \( (-\frac{1}{2}, \frac{1}{2}) \).
Check the endpoints \( x = -\frac{1}{2} \) and \( x = \frac{1}{2} \) by substituting into the series to determine if the series converges at these points. Since the series becomes \( \sum (-1)^n \) or \( \sum 1^n \), analyze their convergence behavior.
Determine absolute convergence by checking if the series \( \sum |(2x)^n| = \sum (2|x|)^n \) converges, which happens when \( |x| < \frac{1}{2} \). For conditional convergence, check if the series converges at endpoints where absolute convergence fails.
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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 is the distance from the center of a power series within which the series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. Finding these involves applying tests like the Ratio Test to determine where the series converges absolutely or conditionally.
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Radius of Convergence
Absolute Convergence
A series converges absolutely if the series of absolute values converges. This means ∑|a_n| converges, ensuring the original series converges regardless of term signs. Absolute convergence guarantees stronger convergence properties and often simplifies analysis of power series.
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07:51Choosing a Convergence Test
Conditional Convergence
Conditional convergence occurs when a series converges, but not absolutely; that is, the series ∑a_n converges while ∑|a_n| diverges. This typically happens at the endpoints of the interval of convergence and requires careful testing, such as the Alternating Series Test, to determine convergence behavior.
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07:51Choosing a Convergence Test
Related Practice
Textbook Question
Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 2 to ∞) ln(n²) / n
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Textbook Question
Telescoping Series
In Exercises 39–44, find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum.
∑ (from n = 1 to ∞) [ (1/n) − (1/(n + 1)) ]
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Textbook Question
Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.
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Textbook Question
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]
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Textbook Question
Using the Ratio Test
In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.
∑(from n=1 to ∞) [(-1)ⁿ (n + 2) / 3ⁿ]