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 (√(n² + n) − n)]
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.133
In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.
aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ
Verified step by step guidance1
Rewrite the given sequence \(a_n = \frac{4^{n+1} + 3^n}{4^n}\) by separating the terms in the numerator over the denominator: \(a_n = \frac{4^{n+1}}{4^n} + \frac{3^n}{4^n}\).
Simplify each term using the properties of exponents: \(\frac{4^{n+1}}{4^n} = 4^{(n+1)-n} = 4^1 = 4\), and \(\frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n\).
Express the sequence in a simpler form: \(a_n = 4 + \left(\frac{3}{4}\right)^n\).
Analyze monotonicity by considering the term \(\left(\frac{3}{4}\right)^n\). Since \(\frac{3}{4} < 1\), this term decreases as \(n\) increases, so \(a_n\) is a decreasing sequence.
Determine boundedness and convergence: since \(\left(\frac{3}{4}\right)^n\) is always positive and approaches 0 as \(n \to \infty\), the sequence is bounded below by 4 and converges to 4.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Monotonic Sequences
A sequence is monotonic if it is either entirely non-increasing or non-decreasing. To determine monotonicity, compare consecutive terms or analyze the general term's behavior as n increases. Monotonic sequences simplify convergence analysis since they move consistently in one direction.
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Introduction to Sequences
Bounded Sequences
A sequence is bounded if all its terms lie within some fixed interval, meaning there exist real numbers M and m such that m ≤ aₙ ≤ M for all n. Boundedness is important because it restricts the sequence's values and is a key condition for convergence in monotonic sequences.
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Convergence of Sequences
A sequence converges if its terms approach a specific finite limit as n approaches infinity. To find the limit, simplify the general term and analyze dominant terms. Convergence indicates the sequence settles to a fixed value, which is crucial for understanding long-term behavior.
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Related Practice
Textbook Question
Find the sum of each series in Exercises 45–52.
∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]
Textbook Question
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
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Textbook Question
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2),a = π/4
Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]
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Textbook Question
Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))
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