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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.8.61
Capitolo 10, Problema 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)

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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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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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06:52
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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Direct 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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Graphs of Logarithmic Functions
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Domanda del libro di testo

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)

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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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Domanda del libro di testo

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

∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!

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Is it possible for a series of positive terms to converge conditionally? Explain.

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21–42. Geometric series Evaluate each geometric series or state that it diverges.  


39.∑ (k = 2 to ∞) (–0.15)ᵏ

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Domanda del libro di testo

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)

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