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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.R.77
Chapter 10, Problem 10.R.77

77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k³⁄⁷

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1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3/7}} \). This is an alternating series because of the factor \( (-1)^{k+1} \), which alternates the sign of each term.
Check for absolute convergence by considering the series of absolute values: \( \sum_{k=1}^{\infty} \left| \frac{(-1)^{k+1}}{k^{3/7}} \right| = \sum_{k=1}^{\infty} \frac{1}{k^{3/7}} \). This is a p-series with \( p = \frac{3}{7} \).
Recall that a p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Since \( \frac{3}{7} < 1 \), the series of absolute values diverges, so the original series does not converge absolutely.
Next, apply the Alternating Series Test to the original series. Check if the terms \( b_k = \frac{1}{k^{3/7}} \) are positive, decreasing, and approach zero as \( k \to \infty \).
Since \( b_k \) is positive, decreasing, and \( \lim_{k \to \infty} b_k = 0 \), the Alternating Series Test confirms that the original 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 Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. Testing absolute convergence often involves comparison or p-series tests.
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Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Alternating series with terms decreasing to zero often exhibit conditional convergence, which requires specific tests like the Alternating Series Test.
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p-Series and Convergence Tests

A p-series ∑1/k^p converges if p > 1 and diverges otherwise. For the given series, the exponent 3/7 is less than 1, so the absolute series diverges. The Alternating Series Test can then be used to check if the original alternating series converges conditionally.
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Related Practice
Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)

Textbook Question

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²

Textbook Question

Sequences versus series

a.Find the limit of the sequence { (−⁴⁄₅)ᵏ }.

1
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Textbook Question

25–26. Recursively defined sequences

The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.


b.Determine the limit of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

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Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)3 / (2 + eᵏ)

Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k⁵ e⁻ᵏ