Textbook Question
What test is advisable if a series involves a factorial term?
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 10.2.55
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(−1)ⁿ / 2ⁿ}
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Identify the general term of the sequence, which is given by \(a_n = \frac{(-1)^n}{2^n}\).
Recall that \((-1)^n\) alternates between \(1\) and \(-1\) as \(n\) increases, causing the numerator to alternate signs.
Note that the denominator \$2^n\( grows exponentially as \)n$ increases, becoming very large.
Consider the behavior of the absolute value of the terms: \(\left|a_n\right| = \frac{1}{2^n}\), which approaches \(0\) as \(n \to \infty\).
Since the numerator only changes sign but the magnitude approaches zero, conclude that the sequence converges to \(0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequence and Limit
A sequence is an ordered list of numbers defined by a specific formula. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding how to find limits helps determine if a sequence converges to a finite value or diverges.
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8:22
Introduction to Sequences
Behavior of Exponential Terms
Exponential terms like 2ⁿ grow rapidly as n increases. When in the denominator, such terms cause the overall fraction to approach zero. Recognizing this behavior is key to evaluating limits involving exponential expressions.
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5:46Graphs of Exponential Functions
Alternating Sequences
An alternating sequence changes sign with each term, often represented by (−1)ⁿ. While the sign alternates, the magnitude may approach zero or another value. Understanding this helps analyze whether the sequence converges or oscillates without settling.
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8:22Introduction to Sequences
Related Practice
Textbook Question
51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
aₙ₊₁ = 4aₙ + 1 a₀ = 1
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Textbook Question
84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b.Use analytical methods to find the limit of the sequence.
aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3
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Textbook Question
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ
Textbook Question
What comparison series would you use with the Comparison Test to determine whether
∑ (k = 1 to ∞) 1 / (k² + 1) converges?
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Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)