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.23
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ
Verified step by step guidance1
Identify the general term of the sequence: \(a_n = \frac{(-1)^n}{0.9^n}\).
Rewrite the denominator to understand its behavior as \(n\) increases: since \(0.9^n = (\frac{9}{10})^n\), it is a positive number less than 1 raised to the power \(n\).
Recall that when a number between 0 and 1 is raised to larger and larger powers, it approaches 0, so \(0.9^n \to 0\) as \(n \to \infty\).
Consider the numerator \((-1)^n\), which oscillates between \(+1\) and \(-1\) depending on whether \(n\) is even or odd.
Combine these observations: since the denominator approaches 0 and the numerator oscillates between \(+1\) and \(-1\), analyze the behavior of the fraction \(\frac{(-1)^n}{0.9^n}\) as \(n\) grows large to determine if the limit exists or diverges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Sequence
The limit of a sequence is 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 to that limit; otherwise, it diverges.
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Introduction to Sequences
Behavior of Exponential Functions
Exponential functions with bases greater than 1 grow without bound, while those with bases between 0 and 1 approach zero as n increases. Understanding this helps determine the dominant behavior of terms like 0.9ⁿ in sequences.
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5:46Graphs of Exponential Functions
Alternating Sequences
An alternating sequence changes sign with each term, often expressed as (–1)ⁿ or (–1)ⁿ⁺¹. Such sequences may oscillate and not have a limit unless the magnitude of terms approaches zero.
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Related Practice
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
Express 0.314141414… as a ratio of two integers.
Textbook Question
Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.
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Textbook Question
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.
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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