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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 10.1.37
Chapter 10, Problem 10.1.37

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. 
aₙ = 1⁄10ⁿ; n = 1, 2, 3, …

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Identify the general term of the sequence given by \(a_{n} = \frac{1}{10^{n}}\) for \(n = 1, 2, 3, \ldots\).
Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula: \(a_{1} = \frac{1}{10^{1}}\), \(a_{2} = \frac{1}{10^{2}}\), \(a_{3} = \frac{1}{10^{3}}\), and \(a_{4} = \frac{1}{10^{4}}\).
Observe the pattern of the terms: each term is one-tenth of the previous term, so the terms get smaller as \(n\) increases.
To determine if the sequence converges, consider the limit of \(a_{n}\) as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{1}{10^{n}}\).
Since \$10^{n}\( grows without bound as \)n$ increases, the terms approach zero, so the sequence converges and its limit is 0.

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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 formula for its nth term. Understanding how to compute the first few terms by substituting values of n helps identify the pattern and behavior of the sequence.
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Limit of a Sequence

The limit of a sequence is the value that the terms approach as n becomes very large. If the terms get arbitrarily close to a fixed number, the sequence converges to that limit; otherwise, it diverges.
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Convergence and Divergence

A sequence converges if its terms approach a finite number as n increases indefinitely. If the terms do not approach any finite value or grow without bound, the sequence diverges. Recognizing this helps in making conjectures about the sequence's behavior.
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Related Practice
Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


21.∑ (k = 0 to ∞) (1/4)ᵏ

Textbook Question

For what values of r does the sequence {rⁿ} converge? Diverge?

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Textbook Question

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 ∞) (−1)ᵏ k³ / √(k⁸ + 1)

Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

Textbook Question

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


∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)

Textbook Question

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 ∞) ((k / (k + 1)) × 2k²)