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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 10 - Sequences and Infinite Series문제 10.1.47
10장, 문제 10.1.47

45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞
 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
aₙ = ⁿ² + n ;n = 1, 2, 3, …

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1
Identify the given sequence formula: \(a_n = n^2 + n\) for \(n = 1, 2, 3, \ldots\).
Create a table of values by substituting \(n\) from 1 to 10 into the formula to find the first ten terms: calculate \(a_1, a_2, \ldots, a_{10}\).
Observe the pattern of the terms in the table to see how the sequence behaves as \(n\) increases.
Analyze the growth of the terms by considering the dominant term \(n^2\) in the formula, which suggests how the sequence behaves for large \(n\).
Based on the behavior of the terms, determine whether the sequence approaches a finite limit or diverges to infinity.

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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.

Sequences and Their Terms

A sequence is an ordered list of numbers defined by a specific formula for its terms. Each term aₙ depends on the index n, which usually starts at 1 and increases by 1. Understanding how to generate terms from the explicit formula is essential for analyzing the sequence's behavior.
추천 영상:
8:22
Introduction to Sequences

Limits of Sequences

The limit of a sequence describes the value that the terms approach as n becomes very large. If the terms get arbitrarily close to a finite number, the sequence converges; otherwise, it diverges. Determining the limit helps understand the long-term behavior of the sequence.
추천 영상:
8:22
Introduction to Sequences

Using Tables and Numerical Approximations

Creating a table of terms allows visualization of the sequence's progression and helps identify patterns or trends. By calculating several terms, one can estimate whether the sequence approaches a finite limit or grows without bound, aiding in conjecturing convergence or divergence.
추천 영상:
6:47
Finding Limits Numerically and Graphically
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교과서 질문

Periodic doses

Suppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.

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교과서 질문

9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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교과서 질문

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) tan(1 / k)

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교과서 질문

61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.


4 + 0.9 + 0.09 + 0.009 + ⋯  

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교과서 질문

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) 2ᵏ k! / kᵏ

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교과서 질문

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))

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