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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 10.2.57
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

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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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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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Root 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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Related Practice
Textbook Question

"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 3aₙ-12; a₁ = 10

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

Textbook Question

8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.


∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4

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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}

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)

Textbook Question

55–70. More sequences

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


{tan⁻¹n / n}

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