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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 10.8.11
Capitolo 10, Problema 10.8.11

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)

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First, write down the general term of the series: \(a_k = \frac{2k^4 + k}{4k^4 - 8k}\).
To analyze convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Simplify the expression by dividing numerator and denominator by the highest power of \(k\) present in the denominator, which is \(k^4\):
\[a_k = \frac{2k^4 + k}{4k^4 - 8k} = \frac{2 + \frac{1}{k^3}}{4 - \frac{8}{k^3}}.\]
Evaluate the limit of \(a_k\) as \(k \to \infty\): \(\lim_{k \to \infty} a_k = \frac{2 + 0}{4 - 0} = \frac{2}{4} = \frac{1}{2}\). Since this limit is not zero, the terms do not approach zero.
Recall the necessary condition for series convergence: if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges. Therefore, conclude that the series 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 as the index grows large, ensuring the sum does not diverge to infinity or oscillate indefinitely.
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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 this limit is a positive finite number, both series either converge or diverge together, making it useful for series with rational expressions.
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Limit Comparison Test

Behavior of Rational Functions for Large k

For large values of k, the dominant terms in the numerator and denominator of a rational function determine its behavior. Simplifying by focusing on highest-degree terms helps approximate the general term, which is essential for applying convergence tests effectively.
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Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)

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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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55–70. More sequences

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


aₙ = (−1)ⁿ ⁿ√n

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13–52. Limits of sequences

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


{(3ⁿ⁺¹ + 3)⁄3ⁿ}

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