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.
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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)}
Verified step by step guidance1
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 \(e^3\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
05:50One-Sided Limits
Related Practice
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(n)}
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ · (k!) / (kᵏ)(Hint: Show that k! / kᵏ ≤ 2 / k², for k ≥ 3.)
Textbook Question
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) (k / (k + 10))ᵏ
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
Textbook Question
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ / k⁰.⁹⁹