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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.R.49
Capitolo 10, Problema 10.R.49

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)

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First, write down the general term of the series: \(a_k = \frac{k^4}{\sqrt{9k^{12} + 2}}\).
To analyze convergence, simplify the expression inside the square root for large \(k\). Notice that \$9k^{12}$ dominates \(2\), so approximate the denominator as \(\sqrt{9k^{12}} = 3k^6\) for large $k$.
Rewrite the term using this approximation: \(a_k \approx \frac{k^4}{3k^6} = \frac{1}{3k^2}\) for large \(k\).
Recognize that the series behaves like \(\sum \frac{1}{k^2}\) for large \(k\), which is a p-series with \(p=2\).
Since a p-series \(\sum \frac{1}{k^p}\) converges if \(p > 1\), conclude that the original series converges by the Comparison Test.

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Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.
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Convergence of an Infinite Series

Comparison Test for Series Convergence

The Comparison Test involves comparing the given series to a second series with known convergence properties. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test is useful when terms resemble simpler series.
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Direct Comparison Test

Asymptotic Behavior and Simplification of Terms

Analyzing the dominant terms in the numerator and denominator for large k helps simplify the general term of the series. This simplification reveals the term's growth rate, which is crucial for applying convergence tests effectively, such as comparing to p-series or geometric series.
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Asymptotes of Hyperbolas
Pratica correlata
Domanda del libro di testo

Express 0.314141414… as a ratio of two integers.

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Domanda del libro di testo

Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.

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Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


e.The sequence aₙ = n² / (n² + 1) converge.

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Domanda del libro di testo

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k√k / k³

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Domanda del libro di testo

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k⁵ e⁻ᵏ

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Domanda del libro di testo

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k³⁄⁷

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