Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)
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 10.2.39
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(1 + (2 / n))ⁿ}
Verified step by step guidance1
Identify the given sequence as \( a_n = \left(1 + \frac{2}{n}\right)^n \).
Recall the important limit definition related to sequences of the form \( \left(1 + \frac{x}{n}\right)^n \), which approaches \( e^x \) as \( n \to \infty \).
In this problem, recognize that \( x = 2 \), so the sequence resembles \( \left(1 + \frac{2}{n}\right)^n \).
Apply the limit property: \( \lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^n = e^2 \).
Conclude that the sequence converges and its limit is \( e^2 \).
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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 becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
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8:22
Introduction to Sequences
Exponential Limit Form (1 + 1/n)^n
The expression (1 + 1/n)^n is a classic limit that approaches the mathematical constant e as n approaches infinity. Variations like (1 + k/n)^n, where k is a constant, approach e raised to the power k, which helps evaluate similar limits.
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Properties of the Number e
The number e (~2.71828) is the base of natural logarithms and arises naturally in limits involving growth processes. It is defined as the limit of (1 + 1/n)^n as n approaches infinity and is fundamental in calculus and exponential functions.
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Related Practice
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)
Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹n / n}
1
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Textbook Question
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 1 to ∞) (k² / (k⁴ + k³ + 1))
Textbook Question
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
2 / 4² + 2 / 5² + 2 / 6² + ⋯