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 / √(k³ᐟ² + k)
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 10.1.17
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
Verified step by step guidance1
Identify the given explicit formula for the sequence: \(a_n = \frac{2^{n+1}}{2^n + 1}\).
Understand that to find the first four terms, you need to substitute \(n = 1, 2, 3, 4\) into the formula.
Calculate the first term by substituting \(n=1\): \(a_1 = \frac{2^{1+1}}{2^1 + 1} = \frac{2^2}{2 + 1}\).
Calculate the second term by substituting \(n=2\): \(a_2 = \frac{2^{2+1}}{2^2 + 1} = \frac{2^3}{4 + 1}\).
Similarly, calculate the third and fourth terms by substituting \(n=3\) and \(n=4\) into the formula, respectively.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and Terms
A sequence is an ordered list of numbers defined by a specific rule or formula. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how to find terms using the explicit formula is essential for generating the sequence.
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Introduction to Sequences
Explicit Formula for Sequences
An explicit formula expresses the nth term of a sequence directly in terms of n, allowing calculation of any term without knowing previous terms. For example, aₙ = (2ⁿ⁺¹) / (2ⁿ + 1) gives a direct way to find the nth term by substituting n.
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Evaluating Exponential Expressions
Evaluating terms in the sequence requires calculating powers of 2, such as 2ⁿ and 2ⁿ⁺¹. Understanding how to compute and simplify exponential expressions is crucial to accurately determine each term's value.
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Related Practice
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)k! / (kᵏ + 3)
Textbook Question
21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)
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 + ⋯