교과서 질문
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
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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 10.1.133
In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.
aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ
검증된 단계별 안내1
Rewrite the given sequence \(a_n = \frac{4^{n+1} + 3^n}{4^n}\) by separating the terms in the numerator over the denominator: \(a_n = \frac{4^{n+1}}{4^n} + \frac{3^n}{4^n}\).
Simplify each term using the properties of exponents: \(\frac{4^{n+1}}{4^n} = 4^{(n+1)-n} = 4^1 = 4\), and \(\frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n\).
Express the sequence in a simpler form: \(a_n = 4 + \left(\frac{3}{4}\right)^n\).
Analyze monotonicity by considering the term \(\left(\frac{3}{4}\right)^n\). Since \(\frac{3}{4} < 1\), this term decreases as \(n\) increases, so \(a_n\) is a decreasing sequence.
Determine boundedness and convergence: since \(\left(\frac{3}{4}\right)^n\) is always positive and approaches 0 as \(n \to \infty\), the sequence is bounded below by 4 and converges to 4.
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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.
Monotonic Sequences
A sequence is monotonic if it is either entirely non-increasing or non-decreasing. To determine monotonicity, compare consecutive terms or analyze the general term's behavior as n increases. Monotonic sequences simplify convergence analysis since they move consistently in one direction.
추천 영상:
8:22
Introduction to Sequences
Bounded Sequences
A sequence is bounded if all its terms lie within some fixed interval, meaning there exist real numbers M and m such that m ≤ aₙ ≤ M for all n. Boundedness is important because it restricts the sequence's values and is a key condition for convergence in monotonic sequences.
추천 영상:
8:22Introduction to Sequences
Convergence of Sequences
A sequence converges if its terms approach a specific finite limit as n approaches infinity. To find the limit, simplify the general term and analyze dominant terms. Convergence indicates the sequence settles to a fixed value, which is crucial for understanding long-term behavior.
추천 영상:
8:22Introduction to Sequences
관련 실천
교과서 질문
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2),a = π/4
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교과서 질문
Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 1 to ∞) 1 / n⁰·²
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교과서 질문
In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.
∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]
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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 ∞) (√n + 1) / (√(n² + 3))
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교과서 질문
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 2 to ∞) [(ln n / n)³]
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