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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 10.4.13
Chapter 10, Problem 10.4.13

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

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Identify the general term of the series: \(a_k = \frac{k}{\ln k}\) for \(k \geq 2\).
Recall the Divergence Test (also known as the Test for Divergence): if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
Compute the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{k}{\ln k}\).
Analyze the behavior of the limit: since \(k\) grows faster than \(\ln k\), the fraction \(\frac{k}{\ln k}\) grows without bound, so the limit does not approach zero.
Conclude that by the Divergence Test, since the limit of \(a_k\) is not zero, the series \(\sum_{k=2}^\infty \frac{k}{\ln k}\) 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. It is a quick way to check divergence but cannot confirm convergence if the limit is zero.
Recommended video:
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Divergence Test (nth Term Test)

Behavior of the General Term

Analyzing the behavior of the general term k / ln(k) as k approaches infinity helps determine if the terms approach zero. Since ln(k) grows slower than k, the term k / ln(k) tends to infinity, indicating the terms do not approach zero.
Recommended video:
05:44
Divergence Test (nth Term Test)

Infinite Series and Convergence

An infinite series converges only if the sum of its terms approaches a finite limit. If the terms do not approach zero, the series cannot converge. Understanding this principle is essential to apply tests like the Divergence Test correctly.
Recommended video:
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Convergence of an Infinite Series
Related Practice
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

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)

Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / ∛k

Textbook Question

55–70. More sequences

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


{nsin³(nπ / 2) / (n + 1)}"

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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

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


∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²