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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

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

문제 10.1.2

10장, 문제 10.1.2

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

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1

Identify the given explicit formula for the sequence: \(a_n = \frac{1}{n}\), where \(n = 1, 2, 3, \ldots\).

Understand that to find the first five terms, you substitute \(n = 1, 2, 3, 4, 5\) into the formula.

Calculate the first term by substituting \(n=1\): \(a_1 = \frac{1}{1}\).

Calculate the second term by substituting \(n=2\): \(a_2 = \frac{1}{2}\).

Continue this process for \(n=3, 4, 5\) to find \(a_3 = \frac{1}{3}\), \(a_4 = \frac{1}{4}\), and \(a_5 = \frac{1}{5}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

2m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sequence Definition

A sequence is an ordered list of numbers defined by a specific rule or formula. Each term in the sequence is identified by its position, usually denoted by n. Understanding how to interpret the formula for aₙ allows you to find any term in the sequence.

추천 영상:

가이드 코스

Introduction to Sequences

Explicit Formula for Sequences

An explicit formula directly expresses the nth term of a sequence as a function of n. For example, aₙ = 1/n means the term at position n is the reciprocal of n. This formula helps compute terms without needing previous terms.

추천 영상:

가이드 코스

Arithmetic Sequences - General Formula

Evaluating Terms of a Sequence

To find specific terms, substitute the term number n into the explicit formula. For aₙ = 1/n, the first five terms are found by plugging in n = 1, 2, 3, 4, and 5, resulting in 1, 1/2, 1/3, 1/4, and 1/5 respectively.

추천 영상:

가이드 코스

Introduction to Sequences

관련 실천

교과서 질문

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)

교과서 질문

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))

교과서 질문

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is

Aₙ = m + mf + ⋯ + mfⁿ⁻¹.

For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.

43.f = 0.25,m = 200 mg

교과서 질문

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-7)ᵏ / k²)

교과서 질문

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

교과서 질문

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹ × k²ᵏ) / (k! × k!)