교과서 질문
9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 0 to ∞) k / (2k + 1)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
10장, 문제 10.7.11
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ
검증된 단계별 안내1
Identify the general term of the series: \(a_k = (-1)^{k+1} \left( \frac{10k^3 + k}{9k^3 + k + 1} \right)^k\).
Since the series has terms raised to the power \(k\), the Root Test is a natural choice. Recall the Root Test uses the limit \(L = \lim_{k \to \infty} \sqrt[k]{|a_k|}\).
Calculate \(\sqrt[k]{|a_k|}\). Because of the absolute value, the \((-1)^{k+1}\) disappears, so \(\sqrt[k]{|a_k|} = \sqrt[k]{\left( \frac{10k^3 + k}{9k^3 + k + 1} \right)^k} = \frac{10k^3 + k}{9k^3 + k + 1}\).
Evaluate the limit \(L = \lim_{k \to \infty} \frac{10k^3 + k}{9k^3 + k + 1}\). To do this, divide numerator and denominator by \(k^3\) to simplify the expression.
Interpret the result of the limit \(L\): if \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the Root Test is inconclusive.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Ratio Test
The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
추천 영상:
가이드 코스
05:35
Ratio Test
Root Test
The Root Test analyzes the nth root of the absolute value of the nth term of a series. If the limit of this root is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. It is especially useful for terms raised to the nth power.
추천 영상:
가이드 코스
07:15Root Test
Absolute Convergence
A series converges absolutely if the series of absolute values of its terms converges. Absolute convergence implies convergence regardless of term signs, which is important when applying tests like the Ratio or Root Test to series with alternating signs.
추천 영상:
가이드 코스
07:51Choosing a Convergence Test
관련 실천
교과서 질문
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)
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교과서 질문
38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.
39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)
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교과서 질문
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) 1 / (klnk)
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교과서 질문
9–15. Geometric sums Evaluate each geometric sum.
∑ k = 0 to 83ᵏ
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교과서 질문
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)
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