Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.2.51

Chapter 10, Problem 10.2.51

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

{nsin(6 / n)}

Verified step by step guidance

Identify the sequence given: \(a_n = n \sin\left(\frac{6}{n}\right)\).

Recall the limit property for sine near zero: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).

Rewrite the sequence term to use this property by setting \(x = \frac{6}{n}\), so \(a_n = n \sin\left(\frac{6}{n}\right) = \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} \times 6\).

Analyze the limit as \(n \to \infty\): since \(\frac{6}{n} \to 0\), use the limit property to find \(\lim_{n \to \infty} \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} = 1\).

Combine the results to conclude that \(\lim_{n \to \infty} a_n = 6 \times 1 = 6\).

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Key Concepts

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

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Behavior of the Sine Function for Small Arguments

For values of x close to zero, sin(x) is approximately equal to x. This linear approximation, sin(x) ≈ x, is useful for evaluating limits involving sine functions where the argument tends to zero.

Limit Laws and Substitution

Limit laws allow the evaluation of limits by breaking complex expressions into simpler parts. Substitution involves replacing variables with their limiting values when the function is continuous, facilitating the calculation of the sequence's limit.

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Related Practice

Textbook Question

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

39.∑ (k = 2 to ∞) (–0.15)ᵏ

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 ∞) (2ᵏ / k⁹⁹)

Textbook Question

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 = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …

Textbook Question

Does a geometric series always have a finite value?

Textbook Question

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

25.∑ (k = 0 to ∞) 0.9ᵏ

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Textbook Question

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

∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]