교과서 질문
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))
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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 10.4.50
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (tanh n) / n²
검증된 단계별 안내1
Identify the general term of the series: \(a_n = \frac{\tanh n}{n^2}\).
Recall that \(\tanh n\) is the hyperbolic tangent function, which approaches 1 as \(n\) becomes very large, so for large \(n\), \(\tanh n \approx 1\).
Compare the given series to the known convergent p-series \(\sum \frac{1}{n^2}\), which converges because \(p=2 > 1\).
Use the Limit Comparison Test by evaluating \(\lim_{n \to \infty} \frac{a_n}{1/n^2} = \lim_{n \to \infty} \tanh n = 1\), which is a finite nonzero number.
Since the limit is finite and nonzero and \(\sum \frac{1}{n^2}\) converges, conclude that the original series \(\sum \frac{\tanh n}{n^2}\) also converges.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of series like ∑ (tanh n) / n².
추천 영상:
가이드 코스
06:52
Convergence of an Infinite Series
Comparison Test for Series
The Comparison Test helps determine convergence by comparing a given series to another series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. This test is useful when dealing with series involving functions like tanh n and polynomial denominators.
추천 영상:
가이드 코스
09:25Direct Comparison Test
Behavior of Hyperbolic Tangent Function (tanh n)
The hyperbolic tangent function, tanh n, approaches 1 as n becomes large. Knowing this limit helps simplify the series terms for large n, allowing us to approximate the series and apply convergence tests effectively, especially when combined with terms like 1/n².
추천 영상:
가이드 코스
05:13Slopes of Tangent Lines
관련 실천
교과서 질문
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4
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교과서 질문
If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.
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교과서 질문
In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [(x + 1)²ⁿ] / 9ⁿ
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교과서 질문
Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) cos²n / n^(3/2)
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교과서 질문
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 8^(1/n)
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