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 / (ln¹⁰ k)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.8.61
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)1 / ln(eᵏ + 1)
Verified step by step guidance1
First, write down the general term of the series: \(a_k = \frac{1}{\ln(e^k + 1)}\).
Simplify the expression inside the logarithm for large \(k\). Since \(e^k\) grows very fast, \(e^k + 1 \approx e^k\), so \(\ln(e^k + 1) \approx \ln(e^k) = k\).
Using this approximation, the general term behaves like \(a_k \approx \frac{1}{k}\) for large \(k\).
Recall that the harmonic series \(\sum \frac{1}{k}\) diverges, so by the Comparison Test or Limit Comparison Test, compare \(a_k\) with \(\frac{1}{k}\) to determine convergence.
Calculate the limit \(\lim_{k \to \infty} \frac{a_k}{1/k}\) to apply the Limit Comparison Test and conclude whether the original series converges or diverges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the sum settles to a finite value or diverges.
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Convergence of an Infinite Series
Comparison Test
The Comparison Test involves comparing the given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it converges; if larger than those of a divergent series, it diverges. This test helps in establishing convergence by bounding.
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09:25Direct Comparison Test
Behavior of Logarithmic Functions in Series
Understanding how logarithmic functions grow is crucial when they appear in series terms. Since ln(e^k + 1) behaves roughly like k for large k, the terms 1/ln(e^k + 1) behave like 1/k, which is a harmonic-type term influencing convergence analysis.
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5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.
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Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)
Textbook Question
Is it possible for a series of positive terms to converge conditionally? Explain.
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
39.∑ (k = 2 to ∞) (–0.15)ᵏ
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 ∞) (2ᵏ / k⁹⁹)