Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.8.41
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)
Verified step by step guidance1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{2^{k}}{3^{k} - 2^{k}} \). We want to determine if this series converges or diverges.
Analyze the general term \( a_k = \frac{2^{k}}{3^{k} - 2^{k}} \). For large \( k \), compare the dominant terms in the denominator to simplify the expression.
Since \( 3^{k} \) grows faster than \( 2^{k} \), for large \( k \), \( 3^{k} - 2^{k} \approx 3^{k} \). So, \( a_k \approx \frac{2^{k}}{3^{k}} = \left( \frac{2}{3} \right)^{k} \).
Use the Comparison Test or Limit Comparison Test by comparing \( a_k \) with the geometric series \( \sum \left( \frac{2}{3} \right)^{k} \), which is a convergent geometric series because \( \left| \frac{2}{3} \right| < 1 \).
Conclude that since \( a_k \) behaves like a convergent geometric series for large \( k \), the original series \( \sum_{k=1}^{\infty} \frac{2^{k}}{3^{k} - 2^{k}} \) converges by the Comparison Test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.
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Convergence of an Infinite Series
Comparison Test
The Comparison Test involves comparing the given series to a known benchmark series with established convergence properties. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps in establishing convergence by bounding the series.
Recommended video:
09:25Direct Comparison Test
Ratio Test
The Ratio Test examines the limit of the ratio of consecutive terms in a series. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly useful for series involving exponential terms, like powers of k, as in the given problem.
Recommended video:
05:35Ratio Test
Related Practice
Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{sinn / 2ⁿ}
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Textbook Question
84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b.Use analytical methods to find the limit of the sequence.
{Use of Tech}aₙ₊₁ = √(2 + aₙ);a₀ = 3
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Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)
Textbook Question
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))
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Textbook Question
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer
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