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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 10.8.7
Chapter 10, Problem 10.8.7

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))

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1
Examine the general term of the series: \(a_k = \frac{k^2}{k^4 + k^3 + 1}\).
To understand the behavior of \(a_k\) for large \(k\), simplify the expression by dividing numerator and denominator by \(k^4\), the highest power in the denominator: \(a_k = \frac{\frac{k^2}{k^4}}{\frac{k^4}{k^4} + \frac{k^3}{k^4} + \frac{1}{k^4}} = \frac{\frac{1}{k^2}}{1 + \frac{1}{k} + \frac{1}{k^4}}\).
As \(k\) approaches infinity, the terms \(\frac{1}{k}\) and \(\frac{1}{k^4}\) approach zero, so \(a_k\) behaves like \(\frac{1/k^2}{1} = \frac{1}{k^2}\).
Since the series behaves like \(\sum \frac{1}{k^2}\) for large \(k\), which is a \(p\)-series with \(p=2 > 1\), the Comparison Test or Limit Comparison Test is appropriate to determine convergence.
Therefore, identify the Comparison Test or Limit Comparison Test as the convergence test to apply, comparing the given series to the convergent \(p\)-series \(\sum \frac{1}{k^2}\).

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Key Concepts

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

Comparison Test

The Comparison Test helps determine convergence by comparing a given series to a second series whose convergence behavior is known. If the terms of the original series are smaller than those of a convergent series, it also converges. Simplifying the terms to a comparable form is often necessary before applying this test.
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Direct Comparison Test

Limit Comparison Test

The Limit Comparison Test involves taking the limit of the ratio of the terms of two series. If the limit is a positive finite number, both series either converge or diverge together. This test is useful when the series terms are complicated but resemble a simpler series asymptotically.
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Limit Comparison Test

Simplifying the General Term

Before applying convergence tests, it is important to simplify or rewrite the general term of the series to identify dominant terms. For example, factoring out the highest powers in numerator and denominator helps reveal the behavior of terms for large indices, making it easier to choose an appropriate test.
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Divergence Test (nth Term Test)
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

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

∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)

Textbook Question

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

∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)

Textbook Question

55–70. More sequences

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


{tan⁻¹n / n}

1
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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

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 + ⋯