Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)
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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.R.1e
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
e.The sequence aₙ = n² / (n² + 1) converge.
Verified step by step guidance1
Identify the sequence given: \(a_n = \frac{n^2}{n^2 + 1}\).
Recall the definition of convergence for a sequence: a sequence \(a_n\) converges to a limit \(L\) if \(\lim_{n \to \infty} a_n = L\) exists and is finite.
To determine if \(a_n\) converges, compute the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{n^2}{n^2 + 1}\).
Divide numerator and denominator by \(n^2\) to simplify the expression: \(\lim_{n \to \infty} \frac{1}{1 + \frac{1}{n^2}}\).
Evaluate the limit by noting that \(\frac{1}{n^2} \to 0\) as \(n \to \infty\), so the limit becomes \(\frac{1}{1 + 0} = 1\), which means the sequence converges to 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequence Convergence
A sequence converges if its terms approach a specific finite value as the index n approaches infinity. This means for any small positive number, the terms eventually get arbitrarily close to that value. Understanding convergence helps determine the long-term behavior of sequences.
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8:22
Introduction to Sequences
Limit of a Sequence
The limit of a sequence is the value that the terms approach as n becomes very large. Calculating the limit often involves simplifying the expression and analyzing dominant terms. If the limit exists and is finite, the sequence converges to that limit.
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8:22Introduction to Sequences
Dominant Term Analysis
When evaluating limits of sequences involving polynomials, the highest degree terms dominate the behavior as n grows large. By comparing the degrees of numerator and denominator, one can simplify the expression to find the limit, which is crucial for determining convergence.
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05:44Divergence Test (nth Term Test)
Related Practice
Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
g.The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.
Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ
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Textbook Question
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.
Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k√k / k³