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 10.4.5
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)
검증된 단계별 안내1
Identify the given series: \( \sum_{n=1}^{\infty} \frac{\cos^{2} n}{n^{3/2}} \). Notice that the terms involve \( \cos^{2} n \) divided by \( n^{3/2} \).
Recall the Direct Comparison Test: To use it, we need to compare the given series to a known benchmark series \( \sum b_n \) where \( b_n \) is positive and whose convergence behavior is known.
Since \( \cos^{2} n \) oscillates between 0 and 1, we know \( 0 \leq \cos^{2} n \leq 1 \) for all \( n \). Therefore, each term of the series satisfies \( 0 \leq \frac{\cos^{2} n}{n^{3/2}} \leq \frac{1}{n^{3/2}} \).
Compare the given series to the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \). Since \( p = \frac{3}{2} > 1 \), this p-series converges.
By the Direct Comparison Test, since \( 0 \leq \frac{\cos^{2} n}{n^{3/2}} \leq \frac{1}{n^{3/2}} \) and \( \sum \frac{1}{n^{3/2}} \) converges, the original series \( \sum \frac{\cos^{2} n}{n^{3/2}} \) also converges.
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주요 개념
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Direct Comparison Test
The Direct Comparison Test determines the convergence or divergence of a series by comparing it to another series with known behavior. If 0 ≤ a_n ≤ b_n for all n and ∑b_n converges, then ∑a_n also converges. Conversely, if a_n ≥ b_n ≥ 0 and ∑b_n diverges, then ∑a_n diverges.
추천 영상:
가이드 코스
09:25
Direct Comparison Test
Behavior of the Cosine Function
The term cos²(n) oscillates between 0 and 1 for all integer n. Since cos²(n) is always non-negative and bounded above by 1, it can be used to compare the given series to a simpler series by replacing cos²(n) with its upper bound.
추천 영상:
가이드 코스
5:53Graph of Sine and Cosine Function
p-Series Test
A p-series ∑ 1/n^p converges if and only if p > 1 and diverges otherwise. In this problem, the denominator n^(3/2) corresponds to p = 3/2, which is greater than 1, so the p-series converges. This fact helps in comparing the given series to a convergent p-series.
추천 영상:
가이드 코스
04:30P-Series and Harmonic Series
관련 실천
교과서 질문
If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.
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교과서 질문
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
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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
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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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 = (2n + 2)! / (2n − 1)!
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