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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 10.5.17
Capitolo 10, Problema 10.5.17

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


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)

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First, write down the general term of the series: \(a_k = \frac{(-k)^3}{3k^3 + 2}\).
Simplify the term \(a_k\): since \((-k)^3 = -k^3\), rewrite \(a_k\) as \(a_k = \frac{-k^3}{3k^3 + 2}\).
Analyze the behavior of \(a_k\) as \(k \to \infty\) by dividing numerator and denominator by \(k^3\): \(a_k = \frac{-1}{3 + \frac{2}{k^3}}\).
Determine the limit of \(a_k\) as \(k \to \infty\): \(\lim_{k \to \infty} a_k = \frac{-1}{3 + 0} = -\frac{1}{3}\).
Since the limit of \(a_k\) is not zero, conclude by the Test for Divergence (also called the nth-term test) that the series \(\sum_{k=1}^\infty a_k\) diverges.

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates without settling.
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Convergence of an Infinite Series

Limit Comparison Test

The Limit Comparison Test compares a given series with a known benchmark series by examining the limit of their term ratios. If the limit is a positive finite number, both series either converge or diverge together, helping to determine the behavior of complex series.
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Behavior of Polynomial Terms in Series

When series terms involve polynomials, the dominant powers in numerator and denominator dictate the term's behavior as k approaches infinity. Simplifying these terms helps identify if the series resembles a p-series or another known type, aiding in applying appropriate convergence tests.
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Domanda del libro di testo

13–52. Limits of sequences

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


{tan⁻¹(10n⁄(10n + 4))}  

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9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)

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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 ∞) ((k / (k + 1)) × 2k²)

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48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) 3ᵏ⁺² / 5ᵏ

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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 ∞) (−7)ᵏ / k!

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Domanda del libro di testo

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³

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