Textbook Question
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
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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 10.2.27
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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Identify the general term of the sequence, which is given by \(a_n = 2^{n+1} 3^{-n}\).
Rewrite the term to express it in a simpler form by separating the powers: \(a_n = 2^{n+1} \times \frac{1}{3^n} = 2 \times 2^n \times 3^{-n}\).
Combine the exponential terms with the same exponent: \(a_n = 2 \times \left( \frac{2}{3} \right)^n\).
Analyze the base of the exponential term \(\left( \frac{2}{3} \right)\) to determine the behavior as \(n\) approaches infinity. Since \(\frac{2}{3} < 1\), the term \(\left( \frac{2}{3} \right)^n\) approaches 0 as \(n \to \infty\).
Conclude that the limit of the sequence \(a_n\) as \(n\) approaches infinity is \(2 \times 0 = 0\).
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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 goes to infinity. If the terms get arbitrarily close to a finite number, the sequence converges; otherwise, it diverges.
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8:22
Introduction to Sequences
Exponential Functions and Their Behavior
Sequences involving exponential terms like 2^(n+1) and 3^(-n) grow or decay depending on the base. Understanding how exponential growth and decay affect the sequence's terms is crucial to determining the limit.
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5:46Graphs of Exponential Functions
Simplifying and Manipulating Sequences
To find limits, it is often necessary to rewrite the sequence in a simpler form, such as combining exponents or factoring terms. This helps reveal dominant terms and makes it easier to analyze the behavior as n approaches infinity.
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8:22Introduction to Sequences
Related Practice
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35.∑ (k = 0 to ∞) 3(–π)^(–k)
74
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√(n² + 1) − n}
45
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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)
65
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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
87
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(√(4n⁴ + 3n))⁄(8n² + 1)}
63
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