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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 10.2.23
Chapter 10, Problem 10.2.23

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.


{(√(4n⁴ + 3n))⁄(8n² + 1)}  

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Identify the given sequence: \(a_n = \frac{\sqrt{4n^4 + 3n}}{8n^2 + 1}\).
To find the limit as \(n \to \infty\), first analyze the dominant terms in the numerator and denominator. The highest power of \(n\) inside the square root is \(n^4\), and in the denominator it is \(n^2\).
Rewrite the numerator by factoring out \(n^4\) inside the square root: \(\sqrt{4n^4 + 3n} = \sqrt{n^4(4 + \frac{3}{n^3})} = n^2 \sqrt{4 + \frac{3}{n^3}}\).
Rewrite the denominator as \(8n^2 + 1 = n^2(8 + \frac{1}{n^2})\).
Express the sequence as \(a_n = \frac{n^2 \sqrt{4 + \frac{3}{n^3}}}{n^2 (8 + \frac{1}{n^2})} = \frac{\sqrt{4 + \frac{3}{n^3}}}{8 + \frac{1}{n^2}}\). Then, take the limit as \(n \to \infty\) by evaluating the limits of the numerator and denominator separately.

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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 finite number, the sequence converges; otherwise, it diverges. Understanding this helps determine the behavior of sequences for large n.
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Asymptotic Behavior and Dominant Terms

When evaluating limits of sequences involving polynomials or roots, focus on the highest-degree terms in numerator and denominator. These dominant terms dictate the growth rate and simplify the expression, making it easier to find the limit as n approaches infinity.
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Simplification Using Algebraic Manipulation

Techniques like factoring, dividing numerator and denominator by the highest power of n, or rationalizing expressions help simplify complex sequences. This process reveals the underlying behavior of the sequence and aids in accurately computing the limit.
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Related Practice
Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)

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Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 2 to ∞) 1 / eᵏ

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. 

{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50;a₀ = 50

Textbook Question

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

∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)

Textbook Question

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

∑ (from k = 1 to ∞)cos(1 / k⁹)

Textbook Question

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

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)