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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 10.5.57
Capitolo 10, Problema 10.5.57

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) 1 / (klnk)

Guida verificata passo dopo passo
1
Identify the series given: \( \sum_{k=2}^{\infty} \frac{1}{k \ln k} \). Notice that the terms are positive and involve a logarithmic function in the denominator.
Recognize that this series resembles a p-series or a series that can be tested using the Integral Test because the terms are positive, continuous, and decreasing for \( k \geq 2 \).
Set up the Integral Test by considering the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \). This integral will help determine the convergence of the series.
Evaluate or analyze the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \) by using the substitution \( u = \ln x \), which implies \( du = \frac{1}{x} dx \). This transforms the integral into \( \int_{\ln 2}^{\infty} \frac{1}{u} \, du \).
Determine the behavior of the integral \( \int_{\ln 2}^{\infty} \frac{1}{u} \, du \). Since this integral diverges (it behaves like the harmonic integral), conclude about the convergence or divergence of the original series based on the Integral Test.

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Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to determine whether the series sums to a finite value or diverges to infinity.
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Convergence of an Infinite Series

Integral Test

The integral test compares a series to an improper integral to determine convergence. If the integral of the corresponding continuous, positive, decreasing function converges, then the series converges; otherwise, it diverges. This test is useful for series involving functions like 1/(k ln k).
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Integral Test

Comparison Test and Limit Comparison Test

These tests compare the given series to a known benchmark series. The comparison test uses inequalities, while the limit comparison test uses limits of term ratios. They help determine convergence by relating the series to simpler, well-understood series such as p-series or harmonic series.
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Limit Comparison Test
Pratica correlata
Domanda del libro di testo

55–70. More sequences

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


{(−1)ⁿ / 2ⁿ}

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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 ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ

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Domanda del libro di testo

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

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Domanda del libro di testo

9–15. Geometric sums Evaluate each geometric sum.


∑ k = 0 to 83ᵏ

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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 ∞) (−1)ᵏ × k / (k³ + 1)

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Domanda del libro di testo

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 ∞) (√k / k − 1)²ᵏ

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