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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.1.33b
Capitolo 10, Problema 10.1.33b

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).
{-5, 5, -5, 5, ......}

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1
Identify the pattern in the given sequence: {-5, 5, -5, 5, ...}. Notice how the terms alternate between -5 and 5.
Recognize that this is an alternating sequence where each term changes sign from the previous term but maintains the same absolute value.
Express the recurrence relation by relating the current term to the previous term. Since the sign alternates, each term is the negative of the previous term. This can be written as: \(a_{n} = -a_{n-1}\).
Specify the initial condition to fully define the sequence. From the given sequence, the first term is \(a_1 = -5\).
State the complete recurrence relation with the initial condition: for \(n \geq 2\), \(a_{n} = -a_{n-1}\), with \(a_1 = -5\).

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Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule. Each number in the sequence is called a term, typically denoted as aₙ, where n indicates the position. Understanding how terms progress helps identify patterns or rules governing the sequence.
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Introduction to Sequences

Recurrence Relation

A recurrence relation expresses each term of a sequence as a function of one or more previous terms. It provides a way to generate the sequence step-by-step, starting from initial term(s). Recognizing the pattern allows formulating this relation to describe the sequence.
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Percorso guidato
04:16
Intro To Related Rates

Initial Conditions

Initial conditions specify the starting term(s) of a sequence, which are necessary to uniquely determine all subsequent terms using the recurrence relation. Without these values, the sequence cannot be fully generated or analyzed.
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Percorso guidato
05:03
Initial Value Problems
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41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.


41. ∑ (k = 1 to ∞) 1 / k⁶

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Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

b. Which of the following series converges faster? Explain.

∑ (k = 2 to ∞) 1 / (k(ln k)²) or ∑ (k = 3 to ∞) 1 / (k(ln k)(ln ln k)²)?

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. A series that converges absolutely must converge.

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71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.


b. Use a geometric series argument with Theorem 10.8.

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.

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72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


b.Find an explicit formula for the terms of the sequence.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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