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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.8.11
Chapter 10, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
Textbook Question

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

aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)

Textbook Question

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 ∞) (−1)ᵏ / √(k³ᐟ² + k)

Textbook Question

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

∑ (from k = 1 to ∞)k! / (kᵏ + 3)

Textbook Question

13–52. Limits of sequences

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


{(1 + (2 / n))ⁿ}

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Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) (k² / (k⁴ + k³ + 1))

Textbook Question

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.

2 / 4² + 2 / 5² + 2 / 6² + ⋯