교과서 질문
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ / √(k³ᐟ² + 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 10.8.59
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)k! / (kᵏ + 3)
검증된 단계별 안내1
First, write down the general term of the series: \(a_k = \frac{k!}{k^k + 3}\).
To determine convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Since \(k^k\) grows very rapidly, compare the growth rates of the numerator \(k!\) and the denominator \(k^k\).
Use the Ratio Test, which involves computing the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\). Substitute \(a_k\) and simplify the expression:
\[L = \lim_{k \to \infty} \frac{(k+1)!}{(k+1)^{k+1} + 3} \cdot \frac{k^k + 3}{k!}.\]
Simplify the factorial and powers, then evaluate the limit \(L\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive and another test should be applied.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Convergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms as the index grows large, ensuring the sum does not diverge to infinity or oscillate indefinitely.
추천 영상:
가이드 코스
06:52
Convergence of an Infinite Series
Ratio Test
The Ratio Test evaluates the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is especially useful for series involving factorials and exponential terms.
추천 영상:
가이드 코스
05:35Ratio Test
Factorials and Exponential Growth Comparison
Factorials (k!) grow faster than exponential functions like k^k for large k, but comparing their growth rates helps determine term behavior. Understanding how factorials compare to powers and exponentials is crucial for applying convergence tests effectively.
추천 영상:
5:22Factorials
관련 실천
교과서 질문
21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2
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교과서 질문
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.
∑ (k = 1 to ∞) 1 / ( (3k + 1)(3k + 4) )
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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 ∞) 20 / (∛k + √k)
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교과서 질문
8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.
∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4
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교과서 질문
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 3 to ∞) 5 / (2 + lnk)
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