Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.4
Finding Terms of a Sequence
Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.
aₙ = 2 + (-1)ⁿ
Verified step by step guidance1
Identify the given formula for the nth term of the sequence: \(a_{n} = 2 + (-1)^{n}\).
Recall that to find specific terms of the sequence, substitute the term number \(n\) into the formula.
Calculate \(a_{1}\) by substituting \(n=1\) into the formula: \(a_{1} = 2 + (-1)^{1}\).
Calculate \(a_{2}\) by substituting \(n=2\) into the formula: \(a_{2} = 2 + (-1)^{2}\).
Similarly, find \(a_{3}\) and \(a_{4}\) by substituting \(n=3\) and \(n=4\) respectively into the formula: \(a_{3} = 2 + (-1)^{3}\) and \(a_{4} = 2 + (-1)^{4}\).
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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 for its terms. Each term is identified by its position n, and the nth term aₙ gives the value at that position. Understanding how to interpret and use the formula for aₙ is essential to find specific terms.
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8:22
Introduction to Sequences
Substitution in Formulas
To find specific terms of a sequence, substitute the term number n into the given formula. This involves replacing n with 1, 2, 3, etc., and simplifying the expression to calculate the corresponding term values accurately.
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04:27Substitution With an Extra Variable
Properties of Exponents and Alternating Signs
The term (-1)ⁿ alternates between -1 and 1 depending on whether n is odd or even. Recognizing this pattern helps determine how the sign affects each term in the sequence, which is crucial for correctly evaluating the formula.
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7:39Introduction to Exponent Rules
Related Practice
Textbook Question
Series with Geometric Terms
In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.
∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]
Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) (1 + 1/n)ⁿ
Textbook Question
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]
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Textbook Question
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (1/n) ∫₁ⁿ (1/x) dx
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Textbook Question
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4