교과서 질문
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [(-1)ⁿ n² e⁻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 10.1.31
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 2 + (0.1)ⁿ
검증된 단계별 안내1
Identify the general term of the sequence: \(a_n = 2 + (0.1)^n\).
Recall that a sequence converges if its terms approach a finite limit as \(n\) approaches infinity.
Analyze the behavior of the term \((0.1)^n\) as \(n \to \infty\). Since \(0.1\) is a number between 0 and 1, \((0.1)^n\) approaches 0 as \(n\) becomes very large.
Use the limit laws to find the limit of the sequence: \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left( 2 + (0.1)^n \right) = 2 + \lim_{n \to \infty} (0.1)^n\).
Since \(\lim_{n \to \infty} (0.1)^n = 0\), conclude that \(\lim_{n \to \infty} a_n = 2 + 0 = 2\), so the sequence converges to 2.
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주요 개념
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Sequence Convergence
A sequence converges if its terms approach a specific finite value as n approaches infinity. This limit is the value the sequence gets arbitrarily close to, indicating stability in the sequence's behavior over time.
추천 영상:
8:22
Introduction to Sequences
Limit of a Sequence
The limit of a sequence {aₙ} is the value L that the terms aₙ approach as n becomes very large. If such a limit exists and is finite, the sequence converges to L; otherwise, it diverges.
추천 영상:
8:22Introduction to Sequences
Behavior of Exponential Terms with Base Between 0 and 1
For a term like (0.1)ⁿ, where the base is between 0 and 1, the value approaches zero as n increases. This property helps determine the limit of sequences involving such terms.
추천 영상:
4:46Solving Exponential Equations Using Like Bases
관련 실천
교과서 질문
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
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교과서 질문
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
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교과서 질문
Convergence or Divergence
Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]
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교과서 질문
In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]
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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 ∞) 1 / (2√n + ³√n)
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