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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 10.7.23
Chapter 10, Problem 10.7.23

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((k / (k + 1)) × 2k²)

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Identify the general term of the series: \(a_k = \left( \frac{k}{k+1} \right) \times 2^{k^2}\).
Recall the Ratio Test: For a series \(\sum a_k\), compute the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.
Compute the ratio \(\frac{a_{k+1}}{a_k} = \frac{\left( \frac{k+1}{k+2} \right) 2^{(k+1)^2}}{\left( \frac{k}{k+1} \right) 2^{k^2}} = \left( \frac{k+1}{k+2} \times \frac{k+1}{k} \right) \times 2^{(k+1)^2 - k^2}\).
Simplify the exponent difference: \((k+1)^2 - k^2 = 2k + 1\), so the ratio becomes \(\left( \frac{(k+1)^2}{k(k+2)} \right) \times 2^{2k+1}\).
Evaluate the limit as \(k \to \infty\) of the ratio. Since \(2^{2k+1}\) grows exponentially, the limit will be infinite, which implies by the Ratio Test that the series diverges.

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Key Concepts

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

Ratio Test

The Ratio Test determines the convergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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Ratio Test

Root Test

The Root Test analyzes convergence by taking the nth root of the absolute value of the nth term of a series. If the limit of this root as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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Root Test

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence regardless of term signs, making tests like the Ratio and Root Tests effective tools for determining the behavior of series with positive or negative terms.
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Choosing a Convergence Test
Related Practice
Textbook Question

35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 

aₙ = 1⁄10ⁿ; n = 1, 2, 3, …

Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

Textbook Question

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


∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)

Textbook Question

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


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)

Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) 3ᵏ⁺² / 5ᵏ

Textbook Question

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

∑ (from k = 1 to ∞) (−7)ᵏ / k!