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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 10.7.45
Chapter 10, Problem 10.7.45

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 / k⁰.⁹⁹

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1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k^{0.99}} \). This is an alternating series because of the factor \( (-1)^k \).
Check if the series converges absolutely by considering the absolute value of the terms: \( \sum_{k=1}^{\infty} \left| \frac{(-1)^k}{k^{0.99}} \right| = \sum_{k=1}^{\infty} \frac{1}{k^{0.99}} \).
Determine whether the series \( \sum_{k=1}^{\infty} \frac{1}{k^{0.99}} \) converges. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Since \( 0.99 < 1 \), this series diverges, so the original series does not converge absolutely.
Apply the Alternating Series Test (Leibniz Test) to the original series. Check two conditions: (1) the terms \( b_k = \frac{1}{k^{0.99}} \) decrease monotonically, and (2) \( \lim_{k \to \infty} b_k = 0 \). Both conditions hold here.
Conclude that since the series converges by the Alternating Series Test but does not converge absolutely, it converges conditionally.

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Key Concepts

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

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the signs of the terms. Testing absolute convergence often involves comparison or p-series tests.
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Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges but does not converge absolutely. This typically occurs in alternating series where the terms decrease in magnitude to zero, but the absolute series diverges. The Alternating Series Test is commonly used to verify this.
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Choosing a Convergence Test

Alternating Series Test

This test determines convergence of series whose terms alternate in sign. If the absolute value of terms decreases monotonically to zero, the series converges. It does not guarantee absolute convergence, only conditional convergence.
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Alternating Series Test
Related Practice
Textbook Question

Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.

Textbook Question

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


∑ (from k = 0 to ∞)3k / ∜(k⁴ + 3)

Textbook Question

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


{n¹⁰ / ln20n}

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Textbook Question

Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{ⁿ√(e³ⁿ⁺⁴)} 

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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 ∞) sin(1 / k) / k²