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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 10.6.57
Chapter 10, Problem 10.6.57

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)

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Identify the given series: \( \sum_{k=1}^{\infty} (-1)^k \cdot \frac{k}{2k + 1} \). This is an alternating series because of the factor \( (-1)^k \).
Check for absolute convergence by considering the absolute value of the terms: \( \sum_{k=1}^{\infty} \left| (-1)^k \cdot \frac{k}{2k + 1} \right| = \sum_{k=1}^{\infty} \frac{k}{2k + 1} \).
Analyze the behavior of the absolute value terms \( \frac{k}{2k + 1} \) as \( k \to \infty \). Determine if the series \( \sum \frac{k}{2k + 1} \) converges or diverges by comparing it to a simpler series.
If the absolute value series diverges, test the original alternating series for conditional convergence using the Alternating Series Test. Verify if the sequence \( b_k = \frac{k}{2k + 1} \) is decreasing and if \( \lim_{k \to \infty} b_k = 0 \).
Based on the results from the absolute convergence test and the Alternating Series Test, conclude whether the series converges absolutely, converges conditionally, or diverges.

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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 ratio 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. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero.
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Alternating Series Test

The Alternating Series Test states that an alternating series ∑(−1)^k b_k converges if the sequence b_k is positive, decreasing, and approaches zero. This test helps determine conditional convergence when absolute convergence fails.
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Related Practice
Textbook Question

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

∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))

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Textbook Question

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

53.0.00952̅ = 0.00952952…

Textbook Question

Growth rates of sequences

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


{n¹⁰⁰⁰ / 2ⁿ}

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Textbook Question

Is it possible for an alternating series to converge absolutely but not conditionally?

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer

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