교과서 질문
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) (k / (k + 10))ᵏ
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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 10.7.9
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) / (k!)
검증된 단계별 안내1
Identify the series given: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k!} \). Since the terms involve factorials, the Ratio Test is often convenient here.
Recall the Ratio Test formula: For the series \( \sum a_k \), compute \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.
Write the general term \( a_k = \frac{(-1)^k}{k!} \). Then, \( a_{k+1} = \frac{(-1)^{k+1}}{(k+1)!} \). Compute the ratio of absolute values: \( \left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{(-1)^{k+1}}{(k+1)!} \cdot \frac{k!}{(-1)^k} \right| = \frac{1}{k+1} \).
Evaluate the limit as \( k \to \infty \): \( L = \lim_{k \to \infty} \frac{1}{k+1} = 0 \). Since \( L < 1 \), the series converges absolutely.
Conclude that by the Ratio Test, the series \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k!} \) converges absolutely because the limit of the ratio of consecutive terms is zero.
비슷한 문제에 대한 검증된 영상 답변:
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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 terms in a series. If the limit of this nth root as n approaches infinity 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
07:15Root Test
Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence implies convergence, and it is a stronger form of convergence that allows the use of tests like the Ratio and Root Tests effectively.
추천 영상:
가이드 코스
07:51Choosing a Convergence Test
관련 실천
교과서 질문
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)
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교과서 질문
21–42. Geometric series Evaluate each geometric series or state that it diverges.
23.∑ (k = 0 to ∞) (–9/10)ᵏ
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교과서 질문
46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
49.0.037̅ = 0.037037…
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교과서 질문
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))
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교과서 질문
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰
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