Textbook Question
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.2.57
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
aₙ = (−1)ⁿ ⁿ√n
Verified step by step guidance1
Identify the given sequence: \(a_n = (-1)^n \sqrt{n}\), where \(\sqrt{n}\) denotes the square root of \(n\).
Recall that the term \((-1)^n\) causes the sequence to alternate signs between positive and negative values as \(n\) increases.
Analyze the behavior of the magnitude of the terms, which is \(\sqrt{n}\). As \(n\) approaches infinity, \(\sqrt{n}\) grows without bound (it increases without limit).
Consider the combined effect: since the magnitude \(\sqrt{n}\) grows larger and the sign alternates, the terms will oscillate between increasingly large positive and negative values.
Conclude that because the terms do not approach a single finite value, the sequence \(a_n\) does not converge and therefore diverges.
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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. If the terms get arbitrarily close to a specific number, the sequence converges; otherwise, it diverges. Understanding limits is essential to analyze the behavior of sequences like aₙ = (−1)ⁿ n^(1/n).
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8:22
Introduction to Sequences
Nth Root and Its Limit
The nth root of n, expressed as n^(1/n), approaches 1 as n becomes very large. This is because the growth of n is tempered by the root, which grows slower, causing the expression to stabilize near 1. Recognizing this helps simplify the sequence's behavior.
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07:15Root Test
Alternating Sequences
Sequences with terms multiplied by (−1)ⁿ alternate in sign between positive and negative values. This oscillation can affect convergence, as the sequence may not settle on a single value. Analyzing the impact of the alternating factor is crucial to determine if the sequence converges or diverges.
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8:22Introduction to Sequences
Related Practice
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(1 + (2 / n))ⁿ}
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