교과서 질문
21–42. Geometric series Evaluate each geometric series or state that it diverges.
41.∑ (k = 1 to ∞) 4 / 12ᵏ
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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 10.1.49
49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.
aₙ = 2 + 2⁻ⁿ;n = 1, 2, 3, …
검증된 단계별 안내1
Identify the sequence given: \(a_n = 2 + 2^{-n}\) for \(n = 1, 2, 3, \ldots\).
Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula:
- \(a_1 = 2 + 2^{-1}\)
- \(a_2 = 2 + 2^{-2}\)
- \(a_3 = 2 + 2^{-3}\)
- \(a_4 = 2 + 2^{-4}\)
Evaluate each term by calculating the powers of 2 with negative exponents (e.g., \(2^{-1} = \frac{1}{2}\), \(2^{-2} = \frac{1}{4}\), etc.) and then add 2 to each result to find the numerical values of the first four terms.
Observe the graph which shows the sequence values plotted against \(n\). Notice that as \(n\) increases, the points approach a horizontal line near \(a_n = 2\).
Based on the formula and the graph, conclude that the plausible limit of the sequence as \(n\) approaches infinity is \(2\), because \(2^{-n}\) approaches 0, making \(a_n\) approach \(2 + 0 = 2\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Sequences and Terms
A sequence is an ordered list of numbers defined by a specific formula for its terms. Each term is identified by its position n, and understanding how to compute the first few terms helps reveal the sequence's behavior. For example, aₙ = 2 + 2⁻ⁿ generates terms approaching a limit as n increases.
추천 영상:
8:22
Introduction to Sequences
Limits of Sequences
The limit of a sequence is the value that the terms approach as n becomes very large. If the terms get closer and closer to a fixed number, that number is the sequence's limit. This concept helps describe long-term behavior, such as aₙ approaching 2 in the given sequence.
추천 영상:
8:22Introduction to Sequences
Interpreting Graphs of Sequences
Graphs visually represent sequence terms as points, showing trends and convergence. By examining the plotted points, one can estimate the limit and verify calculations. In the provided graph, the points approach the horizontal line at aₙ = 2, indicating the sequence's limit.
추천 영상:
8:22Introduction to Sequences
관련 실천
교과서 질문
51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
aₙ₊₁ = 4aₙ + 1 a₀ = 1
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교과서 질문
84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b.Use analytical methods to find the limit of the sequence.
aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3
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교과서 질문
What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?
59
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교과서 질문
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)
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교과서 질문
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰ / ln20n}
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