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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 10.7.49
Chapter 10, Problem 10.7.49

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 / √(k³ᐟ² + k)

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1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \). Notice it is an alternating series because of the factor \( (-1)^k \).
Check for absolute convergence by considering the absolute value of the terms: \( \left| \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \right| = \frac{1}{\sqrt{k^{3/2} + k}} \).
Simplify the denominator inside the square root to understand the behavior for large \( k \): \( \sqrt{k^{3/2} + k} = \sqrt{k^{3/2}(1 + k^{1/2 - 1})} = \sqrt{k^{3/2}} \sqrt{1 + k^{-1/2}} \). Since \( \sqrt{k^{3/2}} = k^{3/4} \), the term behaves like \( \frac{1}{k^{3/4}} \) for large \( k \).
Determine if the series of absolute values \( \sum \frac{1}{k^{3/4}} \) converges. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Here, \( p = \frac{3}{4} < 1 \), so the series does not converge absolutely.
Since the series is alternating, apply the Alternating Series Test: check if the terms \( b_k = \frac{1}{\sqrt{k^{3/2} + k}} \) decrease monotonically to zero. If they do, the series converges conditionally.

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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 series converges, but the series of absolute values diverges. Distinguishing between these helps classify the behavior of alternating series.
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Choosing a Convergence Test

Alternating Series Test

The Alternating Series Test determines convergence of series whose terms alternate in sign. If the absolute value of terms decreases monotonically to zero, the series converges. This test is useful for series with factors like (−1)^k, as in the given problem.
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Alternating Series Test

Comparison and p-Series Tests

Comparison tests involve comparing a given series to a known benchmark series to determine convergence. The p-series test states that ∑ 1/k^p converges if p > 1 and diverges otherwise. Recognizing the dominant term in the denominator helps apply these tests effectively.
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P-Series and Harmonic Series
Related Practice
Textbook Question

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)

Textbook Question

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

∑ (from k = 1 to ∞)k! / (kᵏ + 3)

Textbook Question

21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 2aₙ; a₁ = 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 ∞) 1 / ( (3k + 1)(3k + 4) )

Textbook Question

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

∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)

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.

2 / 4² + 2 / 5² + 2 / 6² + ⋯