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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.7.33
Chapter 10, Problem 10.7.33

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ k (2ᵏ⁺¹ / (9ᵏ − 1))

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First, identify the general term of the series: \(a_k = (-1)^k \cdot k \cdot \frac{2^{k+1}}{9^k - 1}\).
To determine absolute convergence, consider the absolute value of the terms: \(|a_k| = k \cdot \frac{2^{k+1}}{9^k - 1}\).
Analyze the behavior of \(|a_k|\) as \(k\) approaches infinity. Since \$9^k$ grows faster than \(2^{k+1}\), simplify the expression by comparing dominant terms: \(|a_k| \approx k \cdot \frac{2^{k+1}}{9^k}\).
Use the Ratio Test on \(|a_k|\) to check for absolute convergence. Compute the limit \(L = \lim_{k \to \infty} \frac{|a_{k+1}|}{|a_k|}\) and analyze whether \(L < 1\), \(L = 1\), or \(L > 1\).
If the series does not converge absolutely, apply the Alternating Series Test to the original series by checking if the sequence \(b_k = k \cdot \frac{2^{k+1}}{9^k - 1}\) is decreasing and if \(\lim_{k \to \infty} b_k = 0\) to determine conditional convergence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute and Conditional Convergence

Absolute convergence occurs when the series of absolute values converges, ensuring the original series converges regardless of term signs. Conditional convergence happens when the original series converges but the series of absolute values diverges. Distinguishing these helps classify the behavior of alternating or sign-changing series.
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Choosing a Convergence Test

Alternating Series Test

The Alternating Series Test determines convergence for series whose terms alternate in sign. It requires that the absolute value of terms decreases monotonically to zero. If these conditions hold, the series converges, but not necessarily absolutely, indicating possible conditional convergence.
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Alternating Series Test

Ratio Test

The Ratio Test evaluates the limit of the ratio of consecutive terms' absolute values. 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. This test is especially useful for series involving exponential or factorial terms.
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Ratio Test
Related Practice
Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)

Textbook Question

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰

Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)

Textbook Question

9–15. Geometric sums Evaluate each geometric sum.


∑ k = 0 to 83ᵏ

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Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 1 / (√k × e^(√k))