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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.1.43
Chapter 10, Problem 10.1.43

35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 
{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50;a₀ = 50

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Identify the given recursive sequence: \(a_{n+1} = \frac{a_n}{11} + 50\) with initial term \(a_0 = 50\).
Calculate the first four terms by substituting the previous term into the recursive formula: - \(a_1 = \frac{a_0}{11} + 50\) - \(a_2 = \frac{a_1}{11} + 50\) - \(a_3 = \frac{a_2}{11} + 50\) - \(a_4 = \frac{a_3}{11} + 50\).
Write each term explicitly by performing the substitution step-by-step, but do not simplify the numerical values yet.
To analyze convergence, assume the sequence converges to a limit \(L\). Then, by the definition of limit for recursive sequences, set \(L = \frac{L}{11} + 50\).
Solve the equation \(L = \frac{L}{11} + 50\) for \(L\) to find the conjectured limit of the sequence. This will help determine if the sequence converges or diverges.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sequences and Their Terms

A sequence is an ordered list of numbers defined by a specific rule. Each term, denoted as aₙ, depends on its position n. Understanding how to compute initial terms using the given recursive formula is essential to analyze the sequence's behavior.
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Limits and Convergence of Sequences

A sequence converges if its terms approach a specific finite value as n becomes very large. The limit is this value. Recognizing whether a sequence converges or diverges helps in predicting long-term behavior and making conjectures about the limit.
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Recursive Sequences and Fixed Points

Recursive sequences define each term based on previous terms. A fixed point is a value that remains unchanged when plugged into the recursive formula. Finding fixed points helps determine possible limits of the sequence if it converges.
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Related Practice
Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)

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Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 2 to ∞) 1 / eᵏ

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{(√(4n⁴ + 3n))⁄(8n² + 1)}  

Textbook Question

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

∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)

Textbook Question

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

∑ (from k = 1 to ∞)cos(1 / k⁹)

Textbook Question

Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}

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