Textbook Question
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞)ln((2k + 1) / (2k − 1))
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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.R.25b
25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.
b.Determine the limit of each sequence.
25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80
Verified step by step guidance1
Identify the recurrence relation given: \(a_{n+1} = \frac{1}{2} a_n + 8\) with initial value \(a_0 = 80\).
Assuming the sequence converges to a limit \(L\), use the property that as \(n \to \infty\), \(a_n\) and \(a_{n+1}\) both approach \(L\).
Set up the equation for the limit by substituting \(a_n = L\) and \(a_{n+1} = L\) into the recurrence relation: \(L = \frac{1}{2} L + 8\).
Solve the resulting algebraic equation for \(L\) to find the limit of the sequence.
Verify that the sequence is monotonic and bounded to confirm that the limit found is valid.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Recursively Defined Sequences
A recursively defined sequence is one where each term is defined based on one or more previous terms using a recurrence relation. Understanding how to express terms in terms of earlier ones is essential for analyzing the sequence's behavior and finding limits.
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Arithmetic Sequences - Recursive Formula
Monotonicity and Boundedness
A sequence is monotonic if it is either non-increasing or non-decreasing, and bounded if its terms lie within fixed upper and lower limits. These properties guarantee the existence of a limit for the sequence, which is crucial when determining convergence.
Finding Limits of Linear Recurrence Relations
For linear recurrence relations like aₙ₊₁ = r aₙ + c, the limit can be found by setting the limit L equal to its own recurrence: L = rL + c. Solving this equation gives the sequence's limit, assuming |r| < 1 to ensure convergence.
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Related Practice
Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ
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Textbook Question
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏk·e⁻ᵏ
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Textbook Question
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²
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Textbook Question
{Use of Tech} Error in a finite alternating sum
How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?
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Textbook Question
b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?
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