교과서 질문
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (−1)ⁿ / 2ⁿ
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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 10.5.45
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / lnk
검증된 단계별 안내1
Identify the series given: \( \sum_{k=3}^{\infty} \frac{1}{\ln k} \). We want to determine if this series converges or diverges.
Since the terms involve a logarithmic function in the denominator, consider using the Integral Test, which is suitable for positive, continuous, and decreasing functions for \( k \geq 3 \).
Define the function \( f(x) = \frac{1}{\ln x} \) for \( x \geq 3 \). Verify that \( f(x) \) is positive and decreasing on this interval.
Set up the improper integral \( \int_{3}^{\infty} \frac{1}{\ln x} \, dx \) to analyze the behavior of the series using the Integral Test.
Evaluate or analyze the integral \( \int_{3}^{\infty} \frac{1}{\ln x} \, dx \). If the integral diverges, then the series \( \sum_{k=3}^{\infty} \frac{1}{\ln k} \) also diverges; if it converges, then the series converges.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Convergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the partial sums stabilize.
추천 영상:
가이드 코스
06:52
Convergence of an Infinite Series
Comparison Test and Limit Comparison Test
These tests compare the given series to a known benchmark series. If the terms of the given series behave similarly to a convergent or divergent series, the original series shares the same convergence behavior. The limit comparison test uses the limit of the ratio of terms to establish this relationship.
추천 영상:
가이드 코스
07:45Limit Comparison Test
Integral Test
The integral test relates the convergence of a series to the convergence of an improper integral of a related function. If the integral of f(x) from some point to infinity converges, then the series ∑ f(k) also converges, and vice versa. This test is useful when terms involve functions like logarithms.
추천 영상:
가이드 코스
07:25Integral Test
관련 실천
교과서 질문
21–42. Geometric series Evaluate each geometric series or state that it diverges.
37.1 + e/π + e²/π² + e³/π³ + ⋯
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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 ∞) (2k + 1)! / (k!)²
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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^(1/k)
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교과서 질문
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) 5 / (j² + 4)
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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.
61. ∑ (k = 1 to ∞) ln((k + 1) / k)
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