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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 10.4.9
Chapter 10, Problem 10.4.9

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.


∑ (k = 0 to ∞) k / (2k + 1)

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Identify the general term of the series: \(a_k = \frac{k}{2k + 1}\).
Recall the Divergence Test (also known as the nth-term test for divergence): if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
Calculate the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{k}{2k + 1}\).
To find this limit, divide numerator and denominator by \(k\): \(\lim_{k \to \infty} \frac{1}{2 + \frac{1}{k}}\).
Evaluate the limit: as \(k \to \infty\), \(\frac{1}{k} \to 0\), so the limit becomes \(\frac{1}{2}\). Since this limit is not zero, by the Divergence Test, the series diverges.

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Key Concepts

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

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. If the limit is zero, the test is inconclusive, and other methods must be used to determine convergence.
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Divergence Test (nth Term Test)

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index k becomes very large. Evaluating this limit helps determine the behavior of the series' terms, which is essential for applying the Divergence Test.
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Introduction to Sequences

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Understanding whether such a series converges (approaches a finite value) or diverges (does not approach a finite value) is fundamental in calculus, often requiring tests like the Divergence Test.
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Convergence of an Infinite Series
Related Practice
Textbook Question

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

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

Textbook Question

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.


π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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Textbook Question

55–70. More sequences

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


{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

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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.

1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯

Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) k / ln k

Textbook Question

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


∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²