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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.1.2
Capitolo 10, Problema 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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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 \(a_5 = \frac{1}{5}\).

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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.
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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.
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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.
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Introduction to Sequences
Pratica correlata
Domanda del libro di testo

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)

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Domanda del libro di testo

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)))

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Domanda del libro di testo

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

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Domanda del libro di testo

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)

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Domanda del libro di testo

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)

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Domanda del libro di testo

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!)

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