Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√(n² + 1) − n}
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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 10.2.23
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(√(4n⁴ + 3n))⁄(8n² + 1)}
Verified step by step guidance1
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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Introduction to Sequences
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
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
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)cos(1 / k⁹)
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{2ⁿ⁺¹3⁻ⁿ}
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Textbook Question
Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}
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