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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 10.2.59
Chapter 10, Problem 10.2.59

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.


{nsin³(nπ / 2) / (n + 1)}"

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1
First, identify the general term of the sequence: \(a_n = \frac{n \sin^3\left(\frac{n\pi}{2}\right)}{n + 1}\).
Next, analyze the behavior of the sine term \(\sin\left(\frac{n\pi}{2}\right)\) for integer values of \(n\). Recall that \(\sin\left(\frac{n\pi}{2}\right)\) takes on values from the set \({0, \pm 1}\) depending on \(n\) modulo 4.
Determine the pattern of \(\sin^3\left(\frac{n\pi}{2}\right)\) by considering the values of \(\sin\left(\frac{n\pi}{2}\right)\) and then cubing them. Since cubing preserves the sign for \(\pm 1\) and zero remains zero, the sequence will have terms that are \(0\), \(1\), or \(-1\) accordingly.
Rewrite the sequence terms using this pattern, which will simplify \(a_n\) to \(\frac{n \cdot c_n}{n + 1}\) where \(c_n\) is the value of \(\sin^3\left(\frac{n\pi}{2}\right)\) for each \(n\).
Finally, analyze the limit of \(a_n\) as \(n \to \infty\) by considering the behavior of \(\frac{n}{n+1}\) (which approaches 1) multiplied by the oscillating term \(c_n\). Determine if the sequence converges to a limit or diverges based on this oscillation.

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Key Concepts

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

Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index n goes to infinity. Understanding how to evaluate these limits helps determine whether a sequence converges to a finite number or diverges.
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Introduction to Sequences

Behavior of the Sine Function at Multiples of π/2

The sine function has specific values at multiples of π/2: sin(nπ/2) cycles through 0, 1, 0, -1, and repeats. Recognizing this pattern is crucial for simplifying terms like sin³(nπ/2) in the sequence.
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Graph of Sine and Cosine Function

Sequence Divergence and Oscillation

A sequence diverges if it does not approach a single finite limit. Oscillating sequences, which jump between values without settling, are common when trigonometric terms are involved. Identifying oscillation helps conclude divergence.
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Related Practice
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)]

Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / ∛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.

1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯

Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) k / ln k

Textbook Question

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


∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²