Textbook Question
Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.
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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 10.2.41
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{ⁿ√(e³ⁿ⁺⁴)}
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Identify the general term of the sequence: \(a_n = \sqrt[n]{e^{3n + 4}}\).
Rewrite the nth root expression using exponent properties: \(a_n = \left(e^{3n + 4}\right)^{\frac{1}{n}}\).
Simplify the exponent by distributing the \(\frac{1}{n}\): \(a_n = e^{\frac{3n + 4}{n}}\).
Separate the fraction in the exponent: \(a_n = e^{3 + \frac{4}{n}}\).
Analyze the limit as \(n \to \infty\): since \(\frac{4}{n} \to 0\), the limit of \(a_n\) is \(e^3\).
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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 is the value that the terms of the sequence approach as the index goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges. Understanding limits helps determine the long-term behavior of sequences.
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Guided course
8:22
Introduction to Sequences
Properties of Exponential and Root Functions
Exponential functions grow or decay at rates depending on their exponents, while root functions (like nth roots) can be rewritten using fractional exponents. Simplifying expressions involving roots and exponentials often involves rewriting nth roots as powers with exponent 1/n to analyze limits effectively.
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06:21Properties of Functions
Limit Laws and Simplification Techniques
Limit laws allow the separation and simplification of complex expressions into manageable parts. For sequences involving powers and roots, rewriting terms using exponent rules and applying limits to each component helps find the overall limit. Recognizing dominant terms is key to evaluating limits correctly.
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05:50One-Sided Limits
Related Practice
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
41.∑ (k = 1 to ∞) 4 / 12ᵏ
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)
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Textbook Question
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰ / ln20n}
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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 ∞) sin(1 / k) / k²
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Textbook Question
Simplify k! / (k + 2)! for any integer k ≥ 0.
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