Textbook Question
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 = 1 / 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 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 10.6.42
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)]
Verified step by step guidance1
First, identify the general term of the series: \(a_n = (-1)^n (\sqrt{n^2 + n} - n)\).
To analyze absolute convergence, consider the absolute value of the terms: \(|a_n| = \sqrt{n^2 + n} - n\).
Simplify \(|a_n|\) by rationalizing or using algebraic manipulation: multiply numerator and denominator by \(\sqrt{n^2 + n} + n\) to get \(|a_n| = \frac{n}{\sqrt{n^2 + n} + n}\).
Examine the limit of \(|a_n|\) as \(n \to \infty\) to determine if the series of absolute values converges. If it does, the series converges absolutely.
If the series does not converge absolutely, check for conditional convergence by applying the Alternating Series Test: verify if \(a_n\) tends to zero and if the sequence \(|a_n|\) is decreasing for sufficiently large \(n\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Convergence
A series ∑aₙ converges absolutely if the series of absolute values ∑|aₙ| converges. Absolute convergence implies convergence regardless of the sign of terms, and it guarantees that rearranging terms does not affect the sum. Testing absolute convergence often involves comparison or limit comparison tests.
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07:51
Choosing a Convergence Test
Conditional Convergence
A series converges conditionally if it converges but does not converge absolutely. This means ∑aₙ converges, but ∑|aₙ| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero, as described by the Alternating Series Test.
Recommended video:
07:51Choosing a Convergence Test
Alternating Series Test
The Alternating Series Test states that an alternating series ∑(-1)ⁿbₙ converges if the sequence {bₙ} is positive, decreasing, and approaches zero as n→∞. This test helps determine conditional convergence when absolute convergence fails, especially for series with terms involving alternating signs.
Recommended video:
10:54Alternating Series Test
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
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 ∞) (1 − n) / n2ⁿ
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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 = 1 to ∞) [ (x − 1)ⁿ / (n³ 3ⁿ) ]
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