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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 10.6.13
Chapter 10, Problem 10.6.13

11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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Identify the given series: \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k + 1} \). This is an alternating series because of the factor \( (-1)^k \), which causes the terms to alternate in sign.
Recall the Alternating Series Test (Leibniz Test), which states that an alternating series \( \sum (-1)^k a_k \) converges if two conditions are met: (1) the sequence \( a_k \) is positive, decreasing, and (2) \( \lim_{k \to \infty} a_k = 0 \).
Check the sequence \( a_k = \frac{1}{2k + 1} \): it is positive for all \( k \geq 0 \), and as \( k \) increases, \( 2k + 1 \) increases, so \( a_k \) decreases.
Evaluate the limit \( \lim_{k \to \infty} \frac{1}{2k + 1} \). Since the denominator grows without bound, the limit is 0.
Since both conditions of the Alternating Series Test are satisfied, conclude that the series \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k + 1} \) converges.

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Key Concepts

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

Alternating Series Test

The Alternating Series Test determines if a series with terms alternating in sign converges. It requires that the absolute value of the terms decreases monotonically to zero. If these conditions hold, the series converges, though not necessarily absolutely.
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Alternating Series Test

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence involves analyzing term behavior and applying tests like the Alternating Series Test or comparison tests to determine if the sum settles to a finite value.
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Convergence of an Infinite Series

Behavior of the General Term

Examining the general term of a series, such as (−1)^k / (2k + 1), helps assess whether terms decrease in magnitude and approach zero. This behavior is crucial for applying convergence tests and ensuring the series meets necessary criteria for convergence.
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Divergence Test (nth Term 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

For what values of r does the sequence {rⁿ} converge? Diverge?

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Textbook Question

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 k³ / √(k⁸ + 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

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)

Textbook Question

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


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)