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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.6.42
Chapter 10, Problem 10.6.42

Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]

Verified step by step guidance
1
First, identify the general term of the series: \(a_n = (-1)^n (\sqrt{n^2 + n} - n)\).
To analyze absolute convergence, consider the absolute value of the terms: \(|a_n| = \sqrt{n^2 + n} - n\).
Simplify \(|a_n|\) by rationalizing or using algebraic manipulation: multiply numerator and denominator by \(\sqrt{n^2 + n} + n\) to get \(|a_n| = \frac{n}{\sqrt{n^2 + n} + n}\).
Examine the limit of \(|a_n|\) as \(n \to \infty\) to determine if the series of absolute values converges. If it does, the series converges absolutely.
If the series does not converge absolutely, check for conditional convergence by applying the Alternating Series Test: verify if \(a_n\) tends to zero and if the sequence \(|a_n|\) is decreasing for sufficiently large \(n\).

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Key Concepts

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

Absolute Convergence

A series ∑aₙ converges absolutely if the series of absolute values ∑|aₙ| converges. Absolute convergence implies convergence regardless of the sign of terms, and it guarantees that rearranging terms does not affect the sum. Testing absolute convergence often involves comparison or limit comparison tests.
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Conditional Convergence

A series converges conditionally if it converges but does not converge absolutely. This means ∑aₙ converges, but ∑|aₙ| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero, as described by the Alternating Series Test.
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Alternating Series Test

The Alternating Series Test states that an alternating series ∑(-1)ⁿbₙ converges if the sequence {bₙ} is positive, decreasing, and approaches zero as n→∞. This test helps determine conditional convergence when absolute convergence fails, especially for series with terms involving alternating signs.
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Related Practice
Textbook Question

Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;


ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series


a₁/1 + a₂/2 + a₃/3 + …


diverges.

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Textbook Question

Convergent Series

Find the sums of the series in Exercises 19–24.


∑ (from n = 2 to ∞) -2/[n(n+1)]

Textbook Question

Find the sum of each series in Exercises 45–52.

∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]

Textbook Question

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.

aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

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Textbook Question

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))

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Textbook Question

30. b. By differentiating the series in part (a) term by term, show that

Σ(from n=1 to ∞) n / (n + 1)! = 1.