교과서 질문
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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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
10장, 문제 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)]
검증된 단계별 안내1
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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주요 개념
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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.
추천 영상:
가이드 코스
07:51
Choosing a Convergence Test
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.
추천 영상:
가이드 코스
07:51Choosing a Convergence Test
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.
추천 영상:
가이드 코스
10:54Alternating Series Test
관련 실천
교과서 질문
Find the sum of each series in Exercises 45–52.
∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]
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교과서 질문
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
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교과서 질문
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 2 to ∞) [(ln n / n)³]
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교과서 질문
Determining Convergence of Series
Which of the series in Exercises 25–44 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.
∑ (from n = 2 to ∞) 1/[n(ln n)²]
교과서 질문
30. b. By differentiating the series in part (a) term by term, show that
Σ(from n=1 to ∞) n / (n + 1)! = 1.
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